Intermediate Algebra Practice Questions — ACT — Page 4

Question 61Question →

Let the functions $f$ and $g$ be defined by $f(x) = \sqrt{25 - x^2}$ and $g(x) = \frac{1}{\sqrt{x^2 - 9}}$ for all real numbers $x$ where the expressions are defined. What is the number of integers in the domain of the composite function $h(x) = g(f(x))$ ?

Show answer & explanation

Answer: 7

Answer

The number of integers in the domain of the composite function

h(x) = g(f(x))

is 7.

The composite function

h(x) = g(f(x))

is defined when

x

is in the domain of

f

and

f(x)

is in the domain of

g

. The domain of

f(x) = \sqrt{25 - x^2}

is

[-5, 5]

. The domain of

g(u) = \frac{1}{\sqrt{u^2 - 9}}

is

u > 3

or

u < -3

. Substituting

f(x)

for

u

gives

\sqrt{25 - x^2} > 3

, which simplifies to

x^2 < 16

or

-4 < x < 4

. The intersection of

[-5, 5]

and

(-4, 4)

is

(-4, 4)

. The integers in this interval are

-3, -2, -1, 0, 1, 2, 3

, which total 7 integers.

Step-by-Step Solution

1

Determine the domain of the inner function

f(x) = \sqrt{25 - x^2}

.

The domain is

[-5, 5]

.

The term inside the square root must be greater than or equal to zero for the function to yield real values:

25 - x^2 \ge 0

.

2

Determine the domain of the outer function

g(u) = \frac{1}{\sqrt{u^2 - 9}}

.

The domain is

(-\infty, -3) \cup (3, \infty)

.

The expression inside the square root in the denominator must be strictly positive:

u^2 - 9 > 0

.

3

Apply the domain constraint of the outer function to the outputs of the inner function.

-4 < x < 4

.

We require

f(x) > 3

or

f(x) < -3

. Since the range of

f(x)

is non-negative,

f(x) < -3

has no solutions. Thus, we solve

\sqrt{25 - x^2} > 3

, which squares to

25 - x^2 > 9

, or

x^2 < 16

.

4

Find the intersection of the inner function's domain and the composite constraint.

The composite domain is

(-4, 4)

.

The input

x

must satisfy both the domain of

f

(

-5 \le x \le 5

) and the composition constraint (

-4 < x < 4

).

5

List and count the integers within the composite domain

(-4, 4)

.

There are 7 integers.

The integers strictly between

-4

and

4

are

-3, -2, -1, 0, 1, 2, 3

.

Key Concept

Domain of a composite function

Estimated Time:2m 0s

Question 62Question →

If $x$ is a real number, what is the sum of all distinct real solutions to the rational equation $\frac{3}{(x-2)^2} - \frac{2}{x-2} = 1$ ?

3

-3

-2

2

4

Show answer & explanation

Answer: 2

Answer

The sum of all distinct real solutions is 2.

The correct answer is obtained by multiplying the equation by the least common denominator,

(x-2)^2

, which yields the quadratic equation

x^2 - 2x - 3 = 0

. Factoring this equation gives two solutions:

3

and

-1

. Since neither value makes the original denominator equal to zero, both are valid solutions. Summing these values gives a result of

2

.

Step-by-Step Solution

1

Identify the domain constraints of the rational equation.

The denominator

x-2

cannot be zero, so the domain constraint is

x \neq 2

.

Division by zero is undefined.

2

Eliminate the denominators by multiplying the entire equation by the least common denominator, which is

(x-2)^2

.

3 - 2(x-2) = (x-2)^2

This transforms the rational equation into a polynomial equation.

3

Expand both sides of the equation and combine like terms.

3 - 2x + 4 = x^2 - 4x + 4 \implies 7 - 2x = x^2 - 4x + 4

Standard algebraic expansion simplifies the equation into a quadratic form.

4

Rearrange the quadratic equation into standard form

ax^2 + bx + c = 0

.

x^2 - 2x - 3 = 0

Setting the equation to zero allows us to solve for

x

by factoring.

5

Factor the quadratic equation.

(x-3)(x+1) = 0 \implies x = 3

or

x = -1

Factoring finds the values of

x

that satisfy the quadratic equation.

6

Verify if the potential solutions satisfy the domain constraint and calculate their sum.

Both

3

and

-1

are within the domain (

x \neq 2

). The sum is

3 + (-1) = 2

.

Extraneous solutions must be discarded before calculating the final sum.

Key Concept

Solving rational equations by clearing denominators and verifying domain constraints

Question 63Question →

For the function $h(x) = 5 - 3x$ , what is the value of $h(h(2))$ ?

Show answer & explanation

Answer: 8

Answer

8

To find

h(h(2))

, we first calculate the value of the inner function,

h(2) = 5 - 3(2) = 5 - 6 = -1

. We then substitute this result back into the function to evaluate the outer function:

h(-1) = 5 - 3(-1) = 5 + 3 = 8

. Therefore, the correct value is 8.

Step-by-Step Solution

1

Evaluate the inner function

h(2)

h(2) = -1

To evaluate a nested function composition of the form

h(h(x))

, first calculate the value of the inner function at the given input.

2

Evaluate the outer function

h(-1)

using the result from Step 1

h(h(2)) = h(-1) = 8

Substitute the output of the inner function,

-1

, as the new input for the outer function

h(x)

.

Key Concept

Evaluating the composition of a function with itself

Question 64Question →

In the standard $(x, y)$ coordinate plane, a circle is defined by the equation $x^2 + y^2 - 12y + 27 = 0$ . A parabola that opens downward has its vertex at $(0, k)$ and is defined by the equation $y = -x^2 + k$ . If the system of equations consisting of this circle and parabola has exactly three distinct real solution points, what is the value of $k$ ?

Show answer & explanation

Answer: 9

Answer

The value of

k

is 9.

The correct value of

k

is 9 because when

k=9

, the system of equations reduces to a quadratic in

y

with roots

y=9

and

y=4

. Both roots satisfy the real-number constraint

y \leq 9

for the parabola

x^2 = 9-y

, producing three distinct real solutions:

(0, 9)

,

(\sqrt{5}, 4)

, and

(-\sqrt{5}, 4)

.

Step-by-Step Solution

1

Complete the square for the circle's equation.

x^2 + (y-6)^2 = 9

To identify the circle's center at

(0, 6)

and radius

R=3

for geometric interpretation.

2

Express

x^2

in terms of

y

using the parabola's equation.

x^2 = k - y

To substitute into the circle's equation and eliminate the

x

variable.

3

Substitute

x^2

into the circle's equation and simplify.

y^2 - 13y + (k+27) = 0

To create a quadratic equation in

y

representing the y-coordinates of the intersection points.

4

Set

y = k

in the quadratic equation.

k^2 - 12k + 27 = 0

, which factors as

(k-3)(k-9) = 0

An intersection must lie on the y-axis (

x=0

, which means

y=k

) to yield an odd number of intersection points.

5

Verify which candidate value of

k

yields exactly three real solutions.

For

k=3

, the solutions are restricted because

y=10

gives no real

x

value, resulting in only 1 solution. For

k=9

, the roots

y=9

and

y=4

both yield real

x

values, resulting in exactly 3 solutions:

(0, 9)

,

(\sqrt{5}, 4)

, and

(-\sqrt{5}, 4)

.

The algebraic condition for real

x

coordinates is

x^2 = k - y \geq 0

, so we must verify that the roots

y

satisfy

y \leq k

.

Key Concept

Solving systems of non-linear equations algebraically and analyzing the number of real intersection points under coordinate constraints.

Alternative Method

Geometrically, a parabola opening downward with its vertex on the y-axis will intersect a circle centered on the y-axis in exactly three points if and only if its vertex is at the top of the circle and its curvature is less than that of the circle at that point. Completing the square for the circle

x^2 + y^2 - 12y + 27 = 0

gives

x^2 + (y-6)^2 = 9

, which shows the top point of the circle is

(0, 9)

. Thus, the vertex of the downward-opening parabola must be at

(0, 9)

, meaning

k = 9

. We then algebraically verify that this curvature indeed allows two other real intersections.

Estimated Time:3m 0s

Question 65Question →

For all real values of $x$ , what is the real solution to the equation $\sqrt{x+13} - 1 = x$ ?

-4

-3

3

4

11

Show answer & explanation

Answer: 3

Answer

The only real solution to the equation is 3.

Isolating the radical term gives

\sqrt{x+13} = x+1

. Squaring both sides yields

x+13 = x^2 + 2x + 1

. Rearranging into standard quadratic form gives

x^2 + x - 12 = 0

, which factors as

(x+4)(x-3) = 0

. This yields candidate solutions

x = 3

and

x = -4

. Checking these values in the original equation shows that

x = 3

is a valid solution because

\sqrt{3+13} - 1 = 3

, whereas

x = -4

is extraneous because

\sqrt{-4+13} - 1 = 2 \neq -4

.

Step-by-Step Solution

1

Isolate the radical expression on one side of the equation.

\sqrt{x+13} = x+1

This prepares the equation for squaring both sides to eliminate the radical.

2

Square both sides of the equation.

x+13 = (x+1)^2 \implies x+13 = x^2 + 2x + 1

Squaring both sides eliminates the radical and yields a polynomial equation.

3

Move all terms to one side to set the quadratic equation equal to zero.

x^2 + x - 12 = 0

This puts the equation in the standard quadratic form

ax^2 + bx + c = 0

so it can be solved by factoring.

4

Factor the quadratic equation.

(x+4)(x-3) = 0 \implies x = 3

or

x = -4

Factoring allows us to find the potential roots of the quadratic equation.

5

Check both candidate solutions in the original equation to identify any extraneous solutions.

For

x = 3

:

\sqrt{3+13} - 1 = 4 - 1 = 3

(valid). For

x = -4

:

\sqrt{-4+13} - 1 = 3 - 1 = 2 \neq -4

(extraneous).

Squaring both sides of an equation can introduce extraneous roots that must be checked and discarded.

Key Concept

Solving radical equations and checking for extraneous solutions

Estimated Time:1m 30s

Question 66Question →

If $f(x) = 3x - 5$ and $g(x) = (x - 1)^2$ , what is the value of $f(g(4))$ ?

4

12

22

36

40

Show answer & explanation

Answer: 22

Answer

22

To find

f(g(4))

, we evaluate the inner function first:

g(4) = (4 - 1)^2 = 3^2 = 9

. We then evaluate the outer function at this result:

f(9) = 3(9) - 5 = 27 - 5 = 22

.

Step-by-Step Solution

1

Evaluate the inner function

g(x)

at

x = 4

g(4) = (4 - 1)^2 = 3^2 = 9

In function composition

f(g(x))

, the inner function must be evaluated first.

2

Substitute the result from the first step into the outer function

f(x)

f(9) = 3(9) - 5

The output of the inner function becomes the input of the outer function.

3

Calculate the final value of

f(9)

following the order of operations

f(9) = 27 - 5 = 22

Multiplication must be performed before subtraction.

Key Concept

Function Evaluation and Composition

Question 67Question →

Let the complex number $z$ be defined as $z = (4 - 3i)(1 + 2i) + 5i^{14}$ , where $i = \sqrt{-1}$ . What is the real part of $z$ ?

Show answer & explanation

Answer: 5

Answer

The real part of the complex number

z

is

5

.

First, expand the product

(4 - 3i)(1 + 2i)

to get

4 + 8i - 3i - 6i^2

. Replacing

i^2

with

-1

gives

10 + 5i

. Next, simplify

5i^{14}

. Since

i^{14} = (i^4)^3 \cdot i^2 = 1^3 \cdot (-1) = -1

, the term becomes

-5

. Adding the components together gives

z = (10 + 5i) - 5 = 5 + 5i

. The real part of this complex number is

5

.

Step-by-Step Solution

1

Expand the product of the complex binomials

(4 - 3i)(1 + 2i)

10 + 5i

Applying the distributive property gives

4 + 8i - 3i - 6i^2

. Substituting

i^2 = -1

simplifies the expression to

4 + 5i + 6 = 10 + 5i

.

2

Simplify the power of the imaginary unit in

5i^{14}

-5

Since the powers of

i

cycle every 4 terms,

i^{14} = i^{12} \cdot i^2 = 1 \cdot (-1) = -1

. Therefore,

5i^{14} = 5(-1) = -5

.

3

Add the simplified terms together to find

z

5 + 5i

Adding the real and imaginary parts of the terms yields

z = (10 + 5i) + (-5) = 5 + 5i

.

4

Determine the real part of

z

5

A complex number is written in the form

a + bi

, where

a

represents the real part. For

5 + 5i

, the real part is

5

.

Key Concept

Complex multiplication and simplification of powers of the imaginary unit

Question 68Question →

For what values of the real number $p$ is the inequality $-2|p - 4| \geq -10$ true?

p \leq -1

or

p \geq 9

-9 \leq p \leq 1

-1 \leq p \leq 9

p \leq 9

p \geq 9

Show answer & explanation

Answer: $-1 \leq p \leq 9$

Answer

-1 \leq p \leq 9

The correct answer is the inequality showing that

p

is between

-1

and

9

, inclusive. First, divide both sides of the inequality

-2|p - 4| \geq -10

by

-2

and reverse the inequality sign to obtain

|p - 4| \leq 5

. Next, set up the compound inequality

-5 \leq p - 4 \leq 5

. Finally, add

4

to all parts of the inequality to isolate

p

, which yields

-1 \leq p \leq 9

.

Step-by-Step Solution

1

Divide both sides of the inequality

-2|p - 4| \geq -10

by

-2

.

|p - 4| \leq 5

Dividing by a negative number reverses the direction of the inequality sign.

2

Rewrite the absolute value inequality

|p - 4| \leq 5

as a compound inequality.

-5 \leq p - 4 \leq 5

An inequality of the form

|x| \leq a

for

a \geq 0

is equivalent to

-a \leq x \leq a

.

3

Add

4

to all parts of the compound inequality to isolate

p

.

-1 \leq p \leq 9

Adding a constant to all parts of an inequality preserves the inequality relationships and isolates the variable.

Key Concept

Solving absolute value inequalities involving multiplication or division by a negative number.

Question 69Question →

The rational expression $\frac{x^2 - 4}{2x^2 + 5x - 3}$ is undefined for two real values of $x$ . If the smaller value is $a$ and the larger value is $b$ , what are the values of $a$ and $b$ ?

Fill in the blanks below

a =

b =

Show answer & explanation

Answer

The smaller value is

-3

and the larger value is

1/2

(or

0.5

).

A rational expression is undefined when the denominator is equal to zero. To find these values, set the denominator

2x^2 + 5x - 3

equal to

0

and solve for

x

. Factoring the quadratic yields

(2x - 1)(x + 3) = 0

. Setting each factor to zero gives

x = 1/2

and

x = -3

. Since

-3

is less than

1/2

, the smaller value is

-3

and the larger value is

1/2

(or

0.5

).

Step-by-Step Solution

1

Identify the condition that makes a rational expression undefined.

The rational expression is undefined when its denominator is equal to zero:

2x^2 + 5x - 3 = 0

.

Division by zero is undefined in the set of real numbers.

2

Factor the quadratic expression in the denominator.

2x^2 + 5x - 3 = (2x - 1)(x + 3) = 0

.

Factoring allows us to find the roots of the quadratic equation using the zero product property.

3

Solve for the roots of the factored equation.

2x - 1 = 0 \Rightarrow x = 1/2

, and

x + 3 = 0 \Rightarrow x = -3

.

Setting each linear factor to zero determines the values of

x

that make the denominator zero.

4

Assign the values to the variables based on the inequality constraint.

Since

-3 < 1/2

, the smaller value is

a = -3

and the larger value is

b = 1/2

(or

0.5

).

The question specifies that

a

is the smaller value and

b

is the larger value.

Key Concept

Identifying domain restrictions of rational expressions by finding where the denominator is equal to zero.

Estimated Time:1m 30s

Question 70Question →

A quadratic equation of the form $a x^2 + b x + c = 0$ , where $a$ , $b$ , and $c$ are real constants and $a > 0$ , has a discriminant of $37$ . If the sum of the roots of this equation is $5.5$ and the product of the roots is $5.25$ , what is the value of the coefficient $a$ ?

Show answer & explanation

Answer: 2

Answer

The value of the coefficient

a

is

2

.

By applying Vieta's formulas, we can write

b = -5.5a

and

c = 5.25a

. Plugging these into the discriminant formula gives

D = (-5.5a)^2 - 4a(5.25a) = 30.25a^2 - 21a^2 = 9.25a^2

. Setting the discriminant to

37

results in

9.25a^2 = 37 \implies a^2 = 4

. Since the problem specifies

a > 0

, taking the positive square root gives

a = 2

.

Step-by-Step Solution

1

Express the coefficients

b

and

c

in terms of

a

using Vieta's formulas.

b = -5.5a

and

c = 5.25a

The sum of the roots is

-\frac{b}{a}

and the product is

\frac{c}{a}

.

2

Substitute the expressions for

b

and

c

into the discriminant formula

D = b^2 - 4ac

.

D = 9.25a^2

Substituting the terms yields

D = (-5.5a)^2 - 4a(5.25a) = 30.25a^2 - 21a^2 = 9.25a^2

.

3

Equate the discriminant expression to

37

and solve for

a

.

a = 2

Since

D = 37

, we write

9.25a^2 = 37 \implies a^2 = 4

. Because

a

must be positive, we find

a = 2

.

Key Concept

Using the properties of quadratic roots (Vieta's formulas) and the definition of the discriminant to solve for coefficients.

Question 71Question →

For all real numbers $x \geq -5$ , the functions $f$ and $g$ are defined by $f(x) = x^2 - 4x$ and $g(x) = \sqrt{x + 5}$ . If $f(g(k)) = 12$ , what is the real value of $k$ ?

Show answer & explanation

Answer: 31

Answer

The correct answer is 31.

Substituting

g(k) = \sqrt{k+5}

into

f(x) = x^2 - 4x

gives the equation

(\sqrt{k+5})^2 - 4\sqrt{k+5} = 12

. Simplifying and isolating the radical yields

k - 7 = 4\sqrt{k+5}

. Squaring both sides results in

(k-7)^2 = 16(k+5)

, which simplifies to

k^2 - 30k - 31 = 0

. Factoring this equation gives

(k-31)(k+1) = 0

, yielding potential solutions of

k = 31

and

k = -1

. Checking these solutions reveals that

k = -1

is extraneous because

f(g(-1)) = -4 \neq 12

. Therefore, the only valid real solution is

k = 31

.

Step-by-Step Solution

1

Express the composition

f(g(k))

using the given functions.

f(g(k)) = (g(k))^2 - 4(g(k))

To evaluate a composite function, substitute the inner function

g(k)

as the input into the outer function

f

.

2

Substitute

g(k) = \sqrt{k+5}

and set the composite expression equal to 12.

(\sqrt{k+5})^2 - 4\sqrt{k+5} = 12

This sets up the equation to solve for the unknown variable

k

.

3

Isolate the radical term on one side of the equation.

k - 7 = 4\sqrt{k+5}

Simplifying

(\sqrt{k+5})^2

to

k+5

and moving terms helps isolate the radical before squaring.

4

Square both sides of the equation to eliminate the square root.

(k-7)^2 = 16(k+5)

Squaring is the inverse operation of the square root, which removes the radical.

5

Expand both sides and rewrite the equation in standard quadratic form.

k^2 - 30k - 31 = 0

Expanding

(k-7)^2

to

k^2 - 14k + 49

and

16(k+5)

to

16k + 80

, then moving all terms to one side, allows us to solve the resulting quadratic equation.

6

Factor the quadratic equation.

(k-31)(k+1) = 0

Factoring allows us to find the potential roots easily.

7

Solve for the potential values of

k

.

k = 31

or

k = -1

Setting each factor to zero gives the candidate solutions.

8

Verify both potential values in the original equation to check for extraneous solutions.

k = 31

is valid;

k = -1

is extraneous.

Squaring both sides can introduce extraneous solutions. Evaluating

f(g(-1))

yields

-4

, not

12

, while evaluating

f(g(31))

yields

12

.

Key Concept

Function composition involves substituting one function into another, and solving equations with radicals requires checking for extraneous solutions.

Question 72Question →

For all real values of $w$ that satisfy the inequality $4 - |3 - 2w| \leq -1$ , which of the following expressions represents the complete set of possible values of $w$ ?

w \leq -1

or

w \geq 4

-1 \leq w \leq 4

w \leq -4

or

w \geq 1

w \geq 4

w \leq -1

Show answer & explanation

Answer: $w \leq -1$ or $w \geq 4$

Answer

The complete set of possible values is represented by the inequality

w \leq -1

or

w \geq 4

.

The correct answer is the solution set representing

w \leq -1

or

w \geq 4

. Isolating the absolute value expression yields

|3 - 2w| \geq 5

. This splits into two cases:

3 - 2w \geq 5

(which solves to

w \leq -1

after dividing by

-2

and reversing the inequality sign) and

3 - 2w \leq -5

(which solves to

w \geq 4

after dividing by

-2

and reversing the inequality sign). Combining these two cases gives the union

w \leq -1

or

w \geq 4

.

Step-by-Step Solution

1

Isolate the absolute value expression on one side of the inequality.

-|3 - 2w| \leq -5

, which simplifies to

|3 - 2w| \geq 5

after multiplying by

-1

and reversing the inequality sign.

Before splitting an absolute value inequality, the absolute value term must be isolated.

2

Split the absolute value inequality

|3 - 2w| \geq 5

into two separate compound inequalities.

3 - 2w \geq 5

or

3 - 2w \leq -5

An absolute value inequality of the form

|u| \geq c

(where

c > 0

) is equivalent to

u \geq c

or

u \leq -c

.

3

Solve the first inequality:

3 - 2w \geq 5

.

Subtracting

3

from both sides gives

-2w \geq 2

. Dividing both sides by

-2

and reversing the inequality sign gives

w \leq -1

.

Dividing or multiplying an inequality by a negative number requires reversing the direction of the inequality sign.

4

Solve the second inequality:

3 - 2w \leq -5

.

Subtracting

3

from both sides gives

-2w \leq -8

. Dividing both sides by

-2

and reversing the inequality sign gives

w \geq 4

.

Dividing or multiplying an inequality by a negative number requires reversing the direction of the inequality sign.

5

Combine the individual solutions to find the total solution set.

w \leq -1

or

w \geq 4

The solution to a 'greater than or equal to' absolute value inequality is the union of the solutions of the two split cases.

Key Concept

Solving absolute value inequalities with negative variable coefficients by isolating the absolute value term, splitting into cases, and reversing inequality signs when multiplying/dividing by a negative number.

Estimated Time:2m 0s

Question 73Question →

A system of equations consists of the linear equation $y = 2x + 1$ and the quadratic equation $y = x^2 - 2$ . If $(x, y)$ is a solution to this system such that $x > 0$ , what is the value of $y$ ?

Show answer & explanation

Answer: 7

Answer

The correct value of

y

is 7.

Substituting

x = 3

into either equation yields the

y

-value of 7.

Step-by-Step Solution

1

Set the two expressions for

y

equal to each other.

x^2 - 2 = 2x + 1

Since both equations define

y

in terms of

x

, equating them allows us to find the

x

-coordinates of the intersection points.

2

Rearrange the equation into standard quadratic form

ax^2 + bx + c = 0

.

x^2 - 2x - 3 = 0

Subtracting

2x

and

1

from both sides collects all terms on one side of the equation.

3

Factor the quadratic equation.

(x - 3)(x + 1) = 0

Finding two numbers that multiply to

-3

and add to

-2

gives

-3

and

1

, allowing the quadratic to be factored.

4

Solve for

x

and apply the constraint

x > 0

.

x = 3

The factored equation yields solutions of

x = 3

and

x = -1

. The constraint that

x

must be greater than zero means we select

x = 3

.

5

Substitute the

x

-value back into one of the original equations to solve for

y

.

y = 7

Plugging

x = 3

into the linear equation

y = 2x + 1

yields

y = 2(3) + 1 = 7

. Substituting into

y = x^2 - 2

yields

y = 3^2 - 2 = 7

, which confirms the solution.

Key Concept

Solving a system of linear and quadratic equations using substitution.

Question 74Question →

The table below shows some values for the functions $f$ and $g$ .

$x$	$f(x)$	$g(x)$
$-1$	$2$	$3$
$0$	$1$	$-1$
$1$	$-3$	$2$
$2$	$0$	$1$
$3$	$-2$	$0$

What is the value of $g(f(2))$ ?

0

-3

1

-1

2

Show answer & explanation

Answer: $-1$

Answer

The correct answer is

-1

.

To find the value of

g(f(2))

, we begin with the inner function

f(2)

. Looking at the table, when

x = 2

, the value of

f(x)

is

0

. This means

f(2) = 0

. We then substitute this value into the outer function, giving us

g(0)

. Looking at the table again, when

x = 0

, the value of

g(x)

is

-1

. Therefore,

g(f(2)) = -1

.

Step-by-Step Solution

1

Find the value of the inner function

f(2)

from the table.

f(2) = 0

For the composed function

g(f(2))

, the inner function

f

must be evaluated first at the input

2

.

2

Substitute this result into the outer function to evaluate

g(0)

using the table.

g(0) = -1

The output of the inner function becomes the input for the outer function

g

.

Key Concept

Evaluating composite functions using a table of values

Estimated Time:45s

Question 75Question →

Pipe A can fill a water tank in $x$ hours, where $x > 0$ . Pipe B can fill the same water tank in $x + 3$ hours. If both pipes are opened at the same time, they can fill the tank together in exactly 2 hours. What is the value of $x$ ?

0.5

1

2

3

6

Show answer & explanation

Answer: 3

Answer

The value of

x

is 3.

The correct value is 3. The work rate equation for the two pipes is

\frac{1}{x} + \frac{1}{x+3} = \frac{1}{2}

. Combining the fractions on the left gives

\frac{2x+3}{x^2+3x} = \frac{1}{2}

. Cross-multiplying yields

4x + 6 = x^2 + 3x

, which simplifies to the quadratic equation

x^2 - x - 6 = 0

. Factoring this quadratic equation gives

(x-3)(x+2) = 0

, which has the solutions

x = 3

and

x = -2

. Since

x

represents a positive number of hours, the only valid solution is 3.

Step-by-Step Solution

1

Set up the work rate equation based on the individual and combined rates.

\frac{1}{x} + \frac{1}{x+3} = \frac{1}{2}

The sum of the individual rates (in tanks per hour) equals the combined rate of the two pipes.

2

Find a common denominator to combine the fractions on the left side of the equation.

\frac{(x+3) + x}{x(x+3)} = \frac{1}{2} \implies \frac{2x+3}{x^2+3x} = \frac{1}{2}

To add fractions, they must share a common denominator, which is

x(x+3)

.

3

Cross-multiply to eliminate the denominators and form a quadratic equation.

2(2x+3) = x^2+3x \implies 4x + 6 = x^2 + 3x \implies x^2 - x - 6 = 0

Cross-multiplication converts the rational equation into a polynomial equation.

4

Factor the quadratic equation to find the solutions for

x

.

(x-3)(x+2) = 0 \implies x = 3

or

x = -2

Since

x

represents time, it must be positive (

x > 0

), so the only valid solution is

x = 3

.

Key Concept

Solving rational equations in the context of work-rate problems

Question 76Question →

A manufacturer of precision components produces cylindrical rods. The target diameter of the rods is $1.20\text{ centimeters}$ . A rod is classified as Grade A if its actual diameter, $d$ centimeters, satisfies the inequality $|3 - 2.5d| \leq 0.15$ . To be used in a specific high-stress assembly, the rod's diameter must also satisfy the tolerance inequality $|d - 1.22| < 0.04$ . Which of the following inequality expressions represents the complete set of all possible diameters, in centimeters, of rods that qualify as Grade A and are suitable for the assembly?

1.14 \le d \le 1.26

1.14 \le d < 1.26

1.18 < d < 1.26

1.18 < d \le 1.26

1.14 < d < 1.18

Show answer & explanation

Answer: $1.18 < d < 1.26$

Answer

The set of diameters satisfying both conditions is

1.18 < d < 1.26

.

To satisfy both conditions, a rod's diameter must meet the Grade A requirement of

1.14 \le d \le 1.26

and the assembly requirement of

1.18 < d < 1.26

. The intersection of these two intervals is the more restrictive range, which is

1.18 < d < 1.26

. This ensures both inequalities are simultaneously true.

Step-by-Step Solution

1

Solve the first inequality representing Grade A rods:

|3 - 2.5d| \le 0.15

.

1.14 \le d \le 1.26

Rewrite the absolute value inequality as a compound inequality:

-0.15 \le 3 - 2.5d \le 0.15

. Subtract

3

from all parts to get

-3.15 \le -2.5d \le -2.85

. Divide all parts by

-2.5

, reversing the direction of the inequality signs:

1.26 \ge d \ge 1.14

, which simplifies to

1.14 \le d \le 1.26

.

2

Solve the second inequality representing suitability for the assembly:

|d - 1.22| < 0.04

.

1.18 < d < 1.26

Rewrite the absolute value inequality as a compound inequality:

-0.04 < d - 1.22 < 0.04

. Add

1.22

to all parts to isolate

d

:

1.18 < d < 1.26

.

3

Find the intersection of the two solution sets:

[1.14, 1.26] \cap (1.18, 1.26)

.

1.18 < d < 1.26

For a rod to qualify for Grade A and be suitable for the assembly, its diameter must satisfy both conditions. The overlapping range is bounded below by the stricter lower bound of

1.18

(exclusive) and above by the stricter upper bound of

1.26

(exclusive).

Key Concept

Absolute Value Equations and Inequalities

Question 77Question →

For the imaginary unit $i$ , where $i^2 = -1$ , and any integer $n$ , what is the value of the expression $\frac{i^{4n+3} (1 + i)^8}{(1 - i)^6}$ ?

-2

2

-i

-2i

2i

Show answer & explanation

Answer: -2

Answer

-2

To find the value of the expression, we simplify each part. First,

i^{4n+3}

simplifies to

-i

because

i^{4n} = 1

and

i^3 = -i

. Second,

(1+i)^8

is simplified by first squaring the base to get

(1+i)^2 = 2i

, and then raising the result to the fourth power:

(2i)^4 = 16

. Third,

(1-i)^6

is simplified by first squaring the base to get

(1-i)^2 = -2i

, and then cubing the result:

(-2i)^3 = -8i^3 = 8i

. Substituting these back into the expression yields

\frac{-i \cdot 16}{8i} = -2

. Thus, the expression simplifies to

-2

.

Step-by-Step Solution

1

Simplify the term

i^{4n+3}

using the properties of powers of

i

.

i^{4n+3} = -i

Since

i^4 = 1

, we can rewrite

i^{4n+3}

as

(i^4)^n \cdot i^3 = 1^n \cdot (-i) = -i

.

2

Simplify the numerator term

(1+i)^8

.

(1+i)^8 = 16

We can rewrite

(1+i)^8

as

((1+i)^2)^4

. Since

(1+i)^2 = 1 + 2i + i^2 = 2i

, we have

(2i)^4 = 2^4 \cdot i^4 = 16 \cdot 1 = 16

.

3

Simplify the denominator term

(1-i)^6

.

(1-i)^6 = 8i

We can rewrite

(1-i)^6

as

((1-i)^2)^3

. Since

(1-i)^2 = 1 - 2i + i^2 = -2i

, we have

(-2i)^3 = (-2)^3 \cdot i^3 = -8 \cdot (-i) = 8i

.

4

Substitute the simplified components back into the original expression and divide.

\frac{-i \cdot 16}{8i} = -2

Substituting the terms gives

\frac{-i \cdot 16}{8i}

. The common factor of

i

in the numerator and denominator cancels out, and dividing

-16

by

8

yields

-2

.

Key Concept

Simplifying complex expressions involving powers of the imaginary unit and powers of complex binomials.

Estimated Time:1m 30s

Question 78Question →

Let matrix $A = \begin{pmatrix} 2 & 5 \\ -1 & 3 \end{pmatrix}$ and matrix $B = \begin{pmatrix} 4 & -2 \\ 3 & 1 \end{pmatrix}$ . If matrix $C = 2A + 3B$ , what is the value of the entry in the second row and first column of matrix $C$ ?

Show answer & explanation

Answer: 7

Answer

The entry in the second row and first column of matrix

C

is 7.

The entry in the second row, first column of matrix

C

, denoted as

c_{2,1}

, is calculated by multiplying the corresponding entry in matrix

A

(

a_{2,1} = -1

) by 2 and the corresponding entry in matrix

B

(

b_{2,1} = 3

) by 3, and then adding the results:

2(-1) + 3(3) = -2 + 9 = 7

.

Step-by-Step Solution

1

Locate the element in the second row, first column of both matrices

A

and

B

.

The element in the second row, first column of

A

(designated

a_{2,1}

) is

-1

, and the element in the second row, first column of

B

(designated

b_{2,1}

) is

3

.

To find a specific entry in the resulting matrix

C = 2A + 3B

, we perform the scalar multiplication and addition operations on the corresponding entries in the same position of matrices

A

and

B

.

2

Multiply the corresponding elements by their respective scalar factors.

For matrix

A

,

2 \times (-1) = -2

. For matrix

B

,

3 \times 3 = 9

.

Scalar multiplication requires multiplying each individual entry of a matrix by the given scalar factor.

3

Add the two resulting values to find the entry for matrix

C

.

-2 + 9 = 7

Matrix addition is performed by adding the corresponding elements of the matrices.

Key Concept

Matrix scalar multiplication and addition of corresponding elements

Estimated Time:45s

Question 79Question →

A line with a positive slope passes through the point $(0, -4)$ and is tangent to the circle $x^2 + y^2 = 4$ . If this same line is also tangent to the parabola $y = x^2 + k$ , what is the value of the constant $k$ ?

Show answer & explanation

Answer: -3.25

Answer

The constant

k

must be

-3.25

.

By writing the equation of the line passing through

(0, -4)

as

y = mx - 4

and applying the condition that it is tangent to the circle

x^2 + y^2 = 4

, we find the positive slope is

m = \sqrt{3}

. Substituting this tangent line

y = \sqrt{3}x - 4

into the parabola equation

y = x^2 + k

yields the quadratic equation

x^2 - \sqrt{3}x + (k + 4) = 0

. For the line to be tangent to the parabola, the discriminant of this equation must be zero, which gives

3 - 4(k + 4) = 0

, leading to the final value

k = -3.25

.

Step-by-Step Solution

1

Set up the equation of the line passing through

(0, -4)

and substitute it into the equation of the circle.

x^2 + (mx - 4)^2 = 4 \implies (1 + m^2)x^2 - 8mx + 12 = 0

This establishes the quadratic equation representing the intersection points of the line and the circle.

2

Solve for the slope

m

by setting the discriminant of the intersection equation to zero.

(-8m)^2 - 4(1 + m^2)(12) = 0 \implies 64m^2 - 48 - 48m^2 = 0 \implies 16m^2 = 48 \implies m = \sqrt{3}

Since the line is tangent to the circle, the intersection equation must have exactly one real solution, meaning its discriminant is zero. The positive root is selected because the slope is specified as positive.

3

Equate the line

y = \sqrt{3}x - 4

and the parabola

y = x^2 + k

to analyze their intersection.

x^2 - \sqrt{3}x + (k + 4) = 0

This sets up the quadratic equation for the intersection points between the tangent line and the parabola.

4

Solve for

k

by setting the discriminant of this second quadratic equation to zero.

(-\sqrt{3})^2 - 4(1)(k + 4) = 0 \implies 3 - 4k - 16 = 0 \implies -13 - 4k = 0 \implies k = -3.25

For the line to be tangent to the parabola, the system must have exactly one solution, meaning the discriminant of their intersection equation must be zero.

Key Concept

Systems of Linear and Non-Linear Equations

Question 80Question →

Let the functions $f$ and $g$ be defined by $f(x) = \sqrt{3x + 4}$ and $g(x) = x^2 - 2$ . What is the positive value of $x$ for which the composite function value $f(g(x)) = 5$ ?

Show answer & explanation

Answer: 3

Answer

3

The correct answer is 3. We set

f(g(x)) = 5

, which translates to

\sqrt{3g(x) + 4} = 5

. Squaring both sides yields

3g(x) + 4 = 25

. Solving for

g(x)

gives

3g(x) = 21

, which simplifies to

g(x) = 7

. Substituting

g(x) = x^2 - 2

into this equation gives

x^2 - 2 = 7

, which simplifies to

x^2 = 9

. Taking the square root of both sides gives

x = 3

or

x = -3

. Since the question specifies the positive value of

x

, the correct answer is 3.

Step-by-Step Solution

1

Substitute the inner function

g(x)

into the outer function

f(x)

to express

f(g(x)) = 5

\sqrt{3g(x) + 4} = 5

Setting up the initial composite equation to isolate the inner function

g(x)

2

Square both sides of the equation and isolate

g(x)

3g(x) + 4 = 25 \implies 3g(x) = 21 \implies g(x) = 7

Eliminating the radical allows us to determine the required output value of

g(x)

3

Substitute the formula for

g(x)

into the equation and solve for

x

x^2 - 2 = 7 \implies x^2 = 9 \implies x = 3

or

x = -3

Finding the input values of

x

that satisfy the condition for

g(x)

4

Select the positive value from the solution set

x = 3

The question specifically requests the positive value of

x

Key Concept

Evaluating and solving composite function equations by working from the outside function inward

Alternative Method

Instead of solving from the outside in, we can first construct the symbolic expression for

f(g(x))

by substituting

g(x)

directly:

f(g(x)) = \sqrt{3(x^2 - 2) + 4} = \sqrt{3x^2 - 6 + 4} = \sqrt{3x^2 - 2}

. Setting this equal to 5 yields

\sqrt{3x^2 - 2} = 5

. Squaring both sides gives

3x^2 - 2 = 25 \implies 3x^2 = 27 \implies x^2 = 9 \implies x = \pm 3

. The positive solution is 3.

Estimated Time:1m 30s

Intermediate Algebra

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