Function Evaluation, Composition, and Properties Practice Questions — ACT

Question 1Question →

Let the functions $f$ and $g$ be defined by $f(x) = |2x - 5|$ and $g(x) = \sqrt{x + 7}$ for all real numbers in their respective domains. If $a$ is a real number such that $(f \circ g)(a) = 3$ , what is the sum of all possible values of $a$ ?

Show answer & explanation

Answer: 3

Answer

The sum of all possible values of

a

is

3

.

The correct answer is

3

. The composition

(f \circ g)(a) = 3

translates to

f(g(a)) = |2\sqrt{a + 7} - 5| = 3

. This absolute value relation splits into two equations:

2\sqrt{a + 7} - 5 = 3

and

2\sqrt{a + 7} - 5 = -3

. Solving the first equation yields

\sqrt{a + 7} = 4 \Rightarrow a = 9

. Solving the second equation yields

\sqrt{a + 7} = 1 \Rightarrow a = -6

. Since both

9

and

-6

are greater than or equal to

-7

, they are within the domain of the radical function. The sum of these values is

9 + (-6) = 3

.

Step-by-Step Solution

1

Express the composition

(f \circ g)(a)

in terms of

a

.

|2\sqrt{a + 7} - 5| = 3

By definition of function composition,

(f \circ g)(a) = f(g(a))

. Substituting

g(a) = \sqrt{a + 7}

into the expression for

f(x)

yields

f(g(a)) = |2g(a) - 5| = |2\sqrt{a + 7} - 5|

.

2

Set up the two algebraic cases to eliminate the absolute value.

2\sqrt{a + 7} - 5 = 3

or

2\sqrt{a + 7} - 5 = -3

An absolute value equation of the form

|u| = c

(where

c \geq 0

) has two possible cases:

u = c

or

u = -c

.

3

Solve the first equation case for

a

.

a = 9

Adding

5

to both sides of

2\sqrt{a + 7} - 5 = 3

gives

2\sqrt{a + 7} = 8

. Dividing by

2

gives

\sqrt{a + 7} = 4

. Squaring both sides yields

a + 7 = 16

, which gives

a = 9

.

4

Solve the second equation case for

a

.

a = -6

Adding

5

to both sides of

2\sqrt{a + 7} - 5 = -3

gives

2\sqrt{a + 7} = 2

. Dividing by

2

gives

\sqrt{a + 7} = 1

. Squaring both sides yields

a + 7 = 1

, which gives

a = -6

.

5

Verify domain constraints and sum the valid solutions.

3

Both

a = 9

and

a = -6

satisfy the domain requirement for

g(x) = \sqrt{x+7}

, which is

x \geq -7

. The sum of these two valid values is

9 + (-6) = 3

.

Key Concept

Evaluating and solving equations containing composite functions, absolute values, and radical functions.

Alternative Method

Instead of expanding the composition immediately, substitute a temporary variable

u = g(a) = \sqrt{a+7}

. The equation becomes

f(u) = 3 \Rightarrow |2u - 5| = 3

. Solve this simplified absolute value equation to get

2u - 5 = 3 \Rightarrow u = 4

, and

2u - 5 = -3 \Rightarrow u = 1

. Next, substitute back

\sqrt{a+7}

for

u

: solving

\sqrt{a+7} = 4

yields

a = 9

, and solving

\sqrt{a+7} = 1

yields

a = -6

. Summing these two solutions gives

9 + (-6) = 3

.

Estimated Time:2m 0s

Question 2Question →

Let the functions $f$ and $g$ be defined for all real numbers by $f(x) = 2x - 3$ and $g(x) = x + 5$ . What is the value of $f(g(4))$ ?

Show answer & explanation

Answer: 15

Answer

The value of

f(g(4))

is

15

.

To evaluate the composite function

f(g(4))

, the inner function must be evaluated first. Substituting

4

into

g(x) = x + 5

gives

g(4) = 9

. Next, substitute this output of

9

into the outer function

f(x) = 2x - 3

, which gives

f(9) = 2(9) - 3 = 15

.

Step-by-Step Solution

1

Evaluate the inner function

g(4)

g(4) = 9

Substitute

4

for

x

in the definition of

g(x) = x + 5

.

2

Evaluate the outer function

f(x)

at the output of the inner function

f(9) = 15

Substitute

9

(the value of

g(4)

) for

x

in the definition of

f(x) = 2x - 3

.

Key Concept

Function Composition and Evaluation

Estimated Time:45s

Question 3Question →

The table below gives values for the functions $f$ and $g$ at selected values of $x$ :

$x$	$f(x)$	$g(x)$
1	3	5
2	5	4
3	2	1
4	1	2
5	4	3

Based on the table, what is the value of $f(g(3))$ ?

1

2

3

4

5

Show answer & explanation

Answer: 3

Answer

3

To evaluate the composite function

f(g(3))

, we first evaluate the inner function,

g(3)

. Looking at the table, when

x = 3

,

g(3) = 1

. We then substitute this output into the outer function to evaluate

f(1)

. According to the table, when

x = 1

,

f(1) = 3

. Therefore, the value of the composite function is 3.

Step-by-Step Solution

1

Locate the input value of 3 in the table to evaluate the inner function

g(3)

.

g(3) = 1

We must evaluate the inner function first in the composition

f(g(x))

.

2

Substitute the result of

g(3)

into the outer function, giving

f(1)

, and find its value in the table.

f(1) = 3

Evaluating

f(x)

at

x = 1

completes the composition

f(g(3))

.

Key Concept

Evaluating a composite function from a table of values by finding the output of the inner function and using it as the input for the outer function.

Estimated Time:45s

Question 4Question →

For the functions $f(x) = (x - 3)^2 - 4$ and $g(x) = |x + 1|$ , what is the sum of all real values of $x$ for which $f(g(x)) = 5$ ?

-3

-2

4

5

6

Show answer & explanation

Answer: $-3$

Answer

The sum of all real values of

x

for which

f(g(x)) = 5

is

-3

.

To find the sum of all real values of

x

for which

f(g(x)) = 5

, we set up the composite function:

(g(x) - 3)^2 - 4 = 5

. Adding

4

to both sides gives

(g(x) - 3)^2 = 9

. Taking the square root yields two possible cases:

g(x) - 3 = 3

(which means

g(x) = 6

) or

g(x) - 3 = -3

(which means

g(x) = 0

). Next, we substitute

g(x) = |x + 1|

. For the first case,

|x + 1| = 6

gives

x + 1 = 6 \implies x = 5

or

x + 1 = -6 \implies x = -7

. For the second case,

|x + 1| = 0

gives

x + 1 = 0 \implies x = -1

. Summing these three real solutions gives

5 + (-7) + (-1) = -3

.

Step-by-Step Solution

1

Substitute

g(x)

into

f(x)

and set the expression equal to

5

.

f(g(x)) = (g(x) - 3)^2 - 4 = 5

This sets up the composition equation to be solved.

2

Isolate the squared term and solve for

g(x)

by taking the square root of both sides.

(g(x) - 3)^2 = 9

, which gives two cases:

g(x) - 3 = 3 \implies g(x) = 6

or

g(x) - 3 = -3 \implies g(x) = 0

Solving a quadratic equation of the form

u^2 = k

yields two possibilities:

u = \pm\sqrt{k}

.

3

Substitute the expression for

g(x) = |x + 1|

into both cases and solve for

x

.

For

g(x) = 6

,

|x + 1| = 6 \implies x + 1 = 6

or

x + 1 = -6

, yielding

x = 5

or

x = -7

. For

g(x) = 0

,

|x + 1| = 0 \implies x + 1 = 0

, yielding

x = -1

.

Absolute value equations of the form

|v| = c

yield

v = \pm c

if

c > 0

, and

v = 0

if

c = 0

.

4

Calculate the sum of all the real solutions.

5 + (-7) + (-1) = -3

The question asks for the sum of all real values of

x

that satisfy the equation.

Key Concept

Function composition involves substituting one function into another, and solving the resulting equation requires accounting for multiple cases when dealing with quadratic and absolute value expressions.

Question 5Question →

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

Show answer & explanation

Answer: 17

Answer

The value of

g(f(2))

is 17.

To evaluate the composite function

g(f(2))

, we first find the value of the inner function

f(2) = 3(2) - 2 = 4

. Then, we substitute this output as the input for the outer function to get

g(4) = 4^2 + 1 = 17

.

Step-by-Step Solution

1

Substitute

x = 2

into the expression for

f(x)

to evaluate the inner function.

f(2) = 4

In the composition

g(f(2))

, the inner function

f(x)

must be evaluated first at the input value of 2.

2

Substitute the result from Step 1, which is 4, into the expression for

g(x)

.

g(4) = 17

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

g(f(2)) = g(4) = 4^2 + 1 = 17

.

Key Concept

Function Composition

Estimated Time:45s

Question 6Question →

A company's daily profit is modeled by the function $P(x) = 3x - 50$ , where $x$ represents the number of items sold. The number of items sold depends on the number of hours the store is open, $h$ , according to the function $x(h) = 5h$ . What is the daily profit, in dollars, when the store is open for $8$ hours?

-26

40

70

120

170

Show answer & explanation

Answer: 70

Answer

70

Evaluating the composite function in the correct order,

P(x(8))

, yields

70

. First, find the number of items sold when the store is open for 8 hours:

x(8) = 5(8) = 40

. Then, calculate the daily profit by substituting this output into the profit function:

P(40) = 3(40) - 50 = 120 - 50 = 70

.

Step-by-Step Solution

1

Determine the number of items sold,

x

, when the store is open for

8

hours by evaluating the function

x(h) = 5h

at

h = 8

.

x(8) = 5 \times 8 = 40

items.

The number of items sold is the input for the profit function.

2

Substitute the number of items sold (

x = 40

) into the profit function

P(x) = 3x - 50

.

P(40) = 3(40) - 50 = 120 - 50 = 70

dollars.

Evaluating the profit function at the given number of items sold determines the final daily profit.

Key Concept

Evaluating a composite function by finding the output of the inner function and using it as the input for the outer function.

Estimated Time:1m 0s

Question 7Question →

For the functions $f(x) = x^2 - 3x - 1$ and $g(x) = |x - 1|$ , what is the product of all real values of $x$ for which $f(g(x)) = 3$ ?

5

0

-15

-5

-6

Show answer & explanation

Answer: -15

Answer

The product of all real values of

x

is

-15

.

The correct answer is

-15

. By substituting

g(x) = |x - 1|

into

f(x) = x^2 - 3x - 1

, we get

|x - 1|^2 - 3|x - 1| - 1 = 3

. Letting

u = |x - 1|

yields the quadratic

u^2 - 3u - 4 = 0

, which factors into

(u - 4)(u + 1) = 0

. Since

u \geq 0

, we have

u = 4

. Solving

|x - 1| = 4

yields

x = 5

and

x = -3

. The product of these solutions is

5 \times (-3) = -15

.

Step-by-Step Solution

1

Substitute the function

g(x) = |x - 1|

into

f(x) = x^2 - 3x - 1

to form the composite function

f(g(x))

.

f(g(x)) = |x - 1|^2 - 3|x - 1| - 1

To evaluate the composite function before setting it equal to

3

.

2

Set the composite function equal to

3

and substitute

u = |x - 1|

(where

u \geq 0

) to rewrite the equation as a quadratic in terms of

u

.

u^2 - 3u - 1 = 3 \Rightarrow u^2 - 3u - 4 = 0

Using substitution simplifies the absolute value equation into a standard quadratic equation.

3

Factor the quadratic equation

(u - 4)(u + 1) = 0

to solve for

u

, and discard the extraneous solution.

u = 4

or

u = -1

. Since

u = |x - 1| \geq 0

, we discard

u = -1

, leaving

u = 4

.

The absolute value of a real expression must be non-negative, so

u

cannot be

-1

.

4

Solve the absolute value equation

|x - 1| = 4

for all possible real values of

x

.

x - 1 = 4 \Rightarrow x = 5

, and

x - 1 = -4 \Rightarrow x = -3

.

An absolute value equation

|A| = B

(where

B > 0

) yields two scenarios:

A = B

and

A = -B

.

5

Calculate the product of the two real solutions found.

5 \times (-3) = -15

To find the final product of all real values of

x

that satisfy the equation.

Key Concept

Function composition and solving absolute value equations using quadratic substitution

Estimated Time:2m 0s

Question 8Question →

Let the function $f$ be defined by $f(x) = \frac{x - 1}{x + 1}$ for all real numbers $x \neq -1$ . Let $f^1(x) = f(x)$ , and let $f^n(x) = f(f^{n-1}(x))$ for all integers $n \geq 2$ . What is the value of $30 \cdot f^{2026}(3)$ ?

Show answer & explanation

Answer: -10

Answer

The value of the expression is -10.

Evaluating successive iterations of

f(3)

reveals a repeating sequence:

f^1(3) = 1/2

,

f^2(3) = -1/3

,

f^3(3) = -2

,

f^4(3) = 3

, and then

f^5(3) = 1/2

. This indicates a cycle of period 4. Dividing the power 2026 by the period 4 gives a remainder of 2, meaning

f^{2026}(3)

is equal to

f^2(3) = -1/3

. Multiplying this by 30 yields the final answer of -10.

Step-by-Step Solution

1

Calculate the first few compositions of the function evaluated at the given input

x = 3

.

f^1(3) = 1/2

,

f^2(3) = -1/3

,

f^3(3) = -2

, and

f^4(3) = 3

.

To look for a repeating pattern or periodic behavior in the iterated function composition.

2

Identify the period of the repeating cycle.

The cycle has a length of 4, repeating the values

[1/2, -1/3, -2, 3]

.

Since

f^4(3) = 3

, evaluating further iterations will yield the same sequence of values.

3

Use modular arithmetic to find the value of the 2026th composition.

2026 \equiv 2 \pmod 4

, meaning

f^{2026}(3) = f^2(3) = -1/3

.

Since the cycle repeats every 4 iterations, dividing 2026 by 4 yields a remainder of 2, indicating the second value in the sequence.

4

Multiply the computed composition value by the given coefficient.

30 \cdot (-1/3) = -10

.

To find the final value of the expression

30 \cdot f^{2026}(3)

.

Key Concept

Evaluating repeated function compositions using periodicity and modular arithmetic.

Question 9Question →

A weather balloon's altitude in meters after $t$ minutes is given by the function $A(t) = 100 + 50t$ . The air temperature in degrees Celsius at an altitude of $a$ meters is modeled by the function $T(a) = 25 - 0.02a$ . What is the temperature of the air surrounding the balloon, in degrees Celsius, after $2$ minutes?

Show answer & explanation

Answer: 21

Answer

The temperature of the air surrounding the balloon after 2 minutes is 21 degrees Celsius.

The temperature of the air surrounding the balloon after 2 minutes is found by first calculating the balloon's altitude,

A(2) = 100 + 50(2) = 200

meters, and then using this altitude to evaluate the temperature function,

T(200) = 25 - 0.02(200) = 21

.

Step-by-Step Solution

1

Calculate the altitude of the weather balloon at

t = 2

minutes using the function

A(t)

.

A(2) = 200

meters

To find the temperature surrounding the balloon, we must first determine its altitude at the given time of 2 minutes.

2

Substitute the altitude of 200 meters into the temperature function

T(a)

.

T(200) = 21

degrees Celsius

The temperature function depends on the altitude, so evaluating

T(200)

yields the temperature at that height.

Key Concept

Evaluating composite functions in a real-world context.

Question 10Question →

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 11Question →

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 12Question →

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 13Question →

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 14Question →

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 15Question →

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

Question 16Question →

For the functions $f(x) = |x - 2| - 3$ and $g(x) = x^2 - 2x$ , what is the sum of all real values of $x$ for which $g(f(x)) = 0$ ?

2

4

8

12

-8

Show answer & explanation

Answer: 8

Answer

The sum of all real values of

x

is 8.

To solve

g(f(x)) = 0

, we substitute

f(x)

as the variable in

g(x)

, obtaining

(f(x))^2 - 2f(x) = 0

. Factoring this equation yields

f(x)(f(x) - 2) = 0

, giving two cases:

f(x) = 0

or

f(x) = 2

. For the first case,

|x - 2| - 3 = 0

simplifies to

|x - 2| = 3

, which has solutions

x = 5

and

x = -1

. For the second case,

|x - 2| - 3 = 2

simplifies to

|x - 2| = 5

, which has solutions

x = 7

and

x = -3

. Summing these four solutions gives

5 + (-1) + 7 + (-3) = 8

.

Step-by-Step Solution

1

Substitute

f(x)

into

g(x)

to set up the composite equation

g(f(x)) = 0

.

g(f(x)) = (f(x))^2 - 2f(x) = 0

To evaluate a function composition, the output of the inner function

f(x)

becomes the input of the outer function

g(x)

.

2

Factor the quadratic equation in terms of

f(x)

.

f(x)(f(x) - 2) = 0

, which means

f(x) = 0

or

f(x) = 2

.

Factoring allows us to split the quadratic equation into two separate linear cases.

3

Substitute

f(x) = |x - 2| - 3

into the first case,

f(x) = 0

, and solve for

x

.

|x - 2| - 3 = 0 \Rightarrow |x - 2| = 3 \Rightarrow x - 2 = 3

or

x - 2 = -3

, yielding

x = 5

and

x = -1

.

An absolute value equation

|u| = c

has two cases:

u = c

and

u = -c

when

c \geq 0

.

4

Substitute

f(x) = |x - 2| - 3

into the second case,

f(x) = 2

, and solve for

x

.

|x - 2| - 3 = 2 \Rightarrow |x - 2| = 5 \Rightarrow x - 2 = 5

or

x - 2 = -5

, yielding

x = 7

and

x = -3

.

We must solve the second case using both positive and negative scenarios for the absolute value.

5

Calculate the sum of all four real solutions.

5 + (-1) + 7 + (-3) = 8

The question asks for the sum of all real values of

x

that satisfy the composite equation.

Key Concept

Function Composition and Absolute Value Equations

Alternative Method

Instead of expanding the algebraic expressions, substitute a temporary variable

u = f(x)

into

g(u) = 0

to get

u^2 - 2u = 0

, which easily factors to

u = 0

or

u = 2

. Then, substitute

f(x)

back to solve

|x - 2| - 3 = 0

and

|x - 2| - 3 = 2

.

Estimated Time:1m 30s

Question 17Question →

Let the functions $f$ and $g$ be defined for all real numbers by $f(x) = x^2 - 6x + 7$ and $g(x) = |2x - 5|$ . What is the sum of all real values of $x$ for which $f(g(x)) = 14$ ?

3

5

6

8

10

Show answer & explanation

Answer: 5

Answer

The sum of all possible real values of

x

is

5

.

The correct answer is

5

. Setting

f(g(x)) = 14

yields

(g(x))^2 - 6g(x) + 7 = 14

, which simplifies to

(g(x))^2 - 6g(x) - 7 = 0

. Factoring this quadratic gives

g(x) = 7

or

g(x) = -1

. Because

g(x) = |2x - 5|

cannot be negative, we discard the negative case. Solving

|2x - 5| = 7

gives

2x - 5 = 7 \Rightarrow x = 6

and

2x - 5 = -7 \Rightarrow x = -1

. The sum of these values is

6 + (-1) = 5

.

Step-by-Step Solution

1

Set up the composite equation

f(g(x)) = 14

by substituting

g(x)

into

f(x)

.

(g(x))^2 - 6g(x) + 7 = 14

This defines the relation between

g(x)

and the target output value.

2

Rearrange the equation into a standard quadratic form and solve for

g(x)

.

(g(x))^2 - 6g(x) - 7 = 0 \Rightarrow (g(x) - 7)(g(x) + 1) = 0

, so

g(x) = 7

or

g(x) = -1

.

Factoring the quadratic equation gives the possible values for the inner function

g(x)

.

3

Apply the definition of

g(x)

to solve for

x

and discard invalid cases.

Since

g(x) = |2x - 5|

must be non-negative, the case

g(x) = -1

has no real solutions. For the case

g(x) = 7

, we have

|2x - 5| = 7

.

The range of an absolute value function is

[0, \infty)

, making negative outputs impossible.

4

Solve the absolute value equation

|2x - 5| = 7

by setting up both positive and negative cases.

Either

2x - 5 = 7 \Rightarrow 2x = 12 \Rightarrow x = 6

, or

2x - 5 = -7 \Rightarrow 2x = -2 \Rightarrow x = -1

.

An absolute value equation

|u| = c

splits into

u = c

and

u = -c

.

5

Sum the valid solutions for

x

.

6 + (-1) = 5

The question asks for the sum of all real values of

x

that satisfy the equation.

Key Concept

Solving equations involving composite functions and absolute values

Estimated Time:2m 0s

Question 18Question →

For the functions $f(x) = \frac{18}{x - 2}$ and $g(x) = x^2 - 1$ , what is the value of $f(g(3))$ ?

Show answer & explanation

Answer: 3

Answer

The value of the composite function

f(g(3))

is 3.

To evaluate the composite function

f(g(3))

, we start by evaluating the inner function

g(3) = 3^2 - 1 = 8

. We then substitute this result into the outer function, giving

f(8) = \frac{18}{8 - 2} = \frac{18}{6} = 3

.

Step-by-Step Solution

1

Evaluate the inner function

g(3)

8

For any composite function of the form

f(g(x))

, the inner function

g(x)

must be evaluated first at the given input value.

2

Evaluate the outer function

f(8)

3

Substitute the output value of the inner function as the input for the outer function

f(x)

.

Key Concept

Function Composition and Evaluation

Estimated Time:1m 0s

Question 19Question →

The functions $f$ and $g$ are defined for all real numbers by $f(x) = 2x - 5$ and $g(x) = (x + 3)^2$ . What is the value of $f(g(-1))$ ?

15

16

3

27

-28

Show answer & explanation

Answer: 3

Answer

3

To evaluate the composite function

f(g(-1))

, first evaluate the inner function

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

. Then, substitute this result into the outer function to get

f(4) = 2(4) - 5 = 3

.

Step-by-Step Solution

1

Evaluate the inner function

g(x)

at

x = -1

.

g(-1) = 4

In the composition

f(g(-1))

, the inner function must be evaluated first. Substituting

-1

into

g(x) = (x + 3)^2

gives

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

.

2

Evaluate the outer function

f(x)

at the output of the inner function.

f(4) = 3

Substitute the output

g(-1) = 4

into

f(x) = 2x - 5

, which yields

f(4) = 2(4) - 5 = 8 - 5 = 3

.

Key Concept

Function Composition and Evaluation

Estimated Time:1m 0s

Question 20Question →

Consider the function $f$ defined on the domain $[\frac{2}{3}, \infty)$ by the equation $f(x) = 2 + \sqrt{3x - 2}$ . If $f^{-1}$ represents the inverse of $f$ , what is the only real value of $x$ for which $f(x) = f^{-1}(x)$ ?

Show answer & explanation

Answer: 6

Answer

The only real value of

x

for which

f(x) = f^{-1}(x)

is 6.

For a strictly increasing function, the intersection of

f(x)

and

f^{-1}(x)

must occur on the line

y = x

. Equating

f(x) = x

gives

2 + \sqrt{3x - 2} = x

. Isolating the radical term yields

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

. Squaring both sides produces

3x - 2 = x^2 - 4x + 4

, which simplifies to the quadratic equation

x^2 - 7x + 6 = 0

. Factoring this equation gives

(x - 6)(x - 1) = 0

, yielding the solutions

x = 6

and

x = 1

. Substituting these back into the original equation shows that

x = 6

is a valid solution (

2 + \sqrt{16} = 6

), whereas

x = 1

is extraneous because it results in

2 + 1 = 3 \neq 1

. Furthermore, the domain of

f^{-1}

is the range of

f

, which is

[2, \infty)

, meaning

f^{-1}(1)

is undefined. Thus, the only real solution is 6.

Step-by-Step Solution

1

Equate the function to

x

using properties of increasing functions and their inverses.

Since

f(x)

is strictly increasing on its domain

[\frac{2}{3}, \infty)

, the graph of

f(x)

and the graph of its inverse

f^{-1}(x)

can only intersect on the line of symmetry

y = x

. Therefore, the equation

f(x) = f^{-1}(x)

is equivalent to

f(x) = x

.

This simplifies the relation by eliminating the need to solve a high-degree polynomial equation derived from direct composition or substitution.

2

Set up the equation

f(x) = x

and isolate the radical term.

2 + \sqrt{3x - 2} = x \implies \sqrt{3x - 2} = x - 2

Isolating the square root term is a necessary prerequisite step before squaring both sides.

3

Square both sides and rewrite the equation as a standard quadratic equation.

3x - 2 = (x - 2)^2 \implies 3x - 2 = x^2 - 4x + 4 \implies x^2 - 7x + 6 = 0

Squaring eliminates the radical to yield a standard quadratic equation that can be solved analytically.

4

Solve the quadratic equation by factoring.

(x - 6)(x - 1) = 0 \implies x = 6 \text{ or } x = 1

Factoring the quadratic trinomial yields the two candidate values for

x

.

5

Check the candidate solutions in the original equation to eliminate extraneous roots.

For

x = 1

:

2 + \sqrt{3(1)-2} = 3 \neq 1

(extraneous). For

x = 6

:

2 + \sqrt{3(6)-2} = 2 + 4 = 6

(valid). Also, the domain of

f^{-1}

is the range of

f

, which is

[2, \infty)

, thus excluding

x = 1

.

Squaring both sides can introduce extraneous roots. We must verify that the solutions satisfy the original radical equation and fall within the domains of both

f

and

f^{-1}

.

Key Concept

Applying the symmetry of inverse functions about the line

y=x

to solve composition-based equations, while rigorously accounting for domain restrictions and extraneous roots.

Alternative Method

Find the algebraic formula for

f^{-1}(x)

by setting

y = 2 + \sqrt{3x-2}

. Subtracting 2 and squaring both sides gives

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

for

y \geq 2

. Solving for

x

yields

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

for

x \geq 2

. Equating

f(x) = f^{-1}(x)

results in

2 + \sqrt{3x - 2} = \frac{(x - 2)^2 + 2}{3}

. Multiplying by 3 and isolating the radical term gives

3\sqrt{3x - 2} = x^2 - 4x

. Squaring both sides results in a fourth-degree polynomial equation:

9(3x - 2) = (x^2 - 4x)^2 \implies x^4 - 8x^3 + 16x^2 - 27x + 18 = 0

. This factors into

(x - 6)(x - 1)(x^2 - x + 3) = 0

. Since the domain of

f^{-1}(x)

is restricted to

x \geq 2

, the root

x = 1

is rejected, and the quadratic factor

x^2 - x + 3 = 0

has no real roots, leaving

x = 6

as the unique real solution.

Estimated Time:3m 0s

Function Evaluation, Composition, and Properties

Answer

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