Function Evaluation, Composition, and Properties Practice Questions — ACT — Page 2

Question 21Question →

Let the function $h$ be defined by $h(x) = 3x - 5$ , and let the function $g$ be defined by $g(x) = x^2 + c$ for some constant $c$ . If $h(g(2)) = 10$ , what is the value of $c$ ?

Show answer & explanation

Answer: 1

Answer

The value of the constant

c

is

1

.

The correct answer is

1

. Evaluating the inner function

g(2)

gives

2^2 + c = 4 + c

. Applying the outer function

h(x)

to this expression gives

h(4 + c) = 3(4 + c) - 5

. Simplifying this expression results in

12 + 3c - 5 = 7 + 3c

. We are given that this value equals

10

, so

7 + 3c = 10

. Subtracting 7 from both sides gives

3c = 3

, which yields

c = 1

.

Step-by-Step Solution

1

Evaluate

g(2)

in terms of

c

g(2) = 4 + c

We substitute

x = 2

into the definition

g(x) = x^2 + c

to get

2^2 + c = 4 + c

.

2

Substitute

g(2)

into the definition of

h(x)

to express

h(g(2))

h(g(2)) = 3(4 + c) - 5

Since the composition is

h(g(2))

, we evaluate the function

h

at the input value

g(2) = 4 + c

.

3

Set

h(g(2)) = 10

and solve the linear equation for

c

c = 1

Expanding and simplifying the equation

3(4 + c) - 5 = 10

gives

12 + 3c - 5 = 10

, which reduces to

7 + 3c = 10

. Subtracting 7 gives

3c = 3

, so

c = 1

.

Key Concept

Function Composition and Parameter Evaluation

Estimated Time:1m 15s

Question 22Question →

For real numbers $x$ where the functions are defined, let $f(x) = \frac{x}{x^2 - 4}$ and $g(x) = \frac{x+3}{x-1}$ . What is the set of all real numbers $x$ for which the composite function $h(x) = f(g(x))$ is undefined?

\left\{1\right\}

\left\{-2, 1, 2\right\}

\left\{1, 5\right\}

\left\{-\frac{1}{3}, 1, 5\right\}

\left\{-\frac{1}{3}, -2, 1, 2, 5\right\}

Show answer & explanation

Answer: $\left\{-\frac{1}{3}, 1, 5\right\}$

Answer

The set containing negative one-third, one, and five

The correct answer is the set containing negative one-third, one, and five. The composite function

f(g(x))

is undefined when the inner function

g(x)

is undefined (which occurs at

x = 1

) or when the output of

g(x)

equals a value that makes the outer function

f(x)

undefined. The outer function

f(x)

is undefined when

x^2 - 4 = 0

, which gives

x = \pm 2

. Setting

g(x) = 2

leads to

x = 5

, and setting

g(x) = -2

leads to

x = -\frac{1}{3}

. Collecting all these values gives the complete set of restrictions.

Step-by-Step Solution

1

Determine the domain restriction of the inner function

x \neq 1

For the composite function

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

to be defined, the input

x

must first be in the domain of the inner function

g(x) = \frac{x+3}{x-1}

. Thus, the denominator cannot be zero, which gives

x - 1 \neq 0 \implies x \neq 1

.

2

Determine the restricted inputs for the outer function

u \neq 2

and

u \neq -2

The outer function

f(u) = \frac{u}{u^2 - 4}

is undefined when its denominator is zero. Solving

u^2 - 4 = 0

yields

u^2 = 4 \implies u = 2

or

u = -2

. Therefore, the output of the inner function,

g(x)

, cannot equal

2

or

-2

.

3

Solve the equation for the first restricted value of the outer function

x = 5

Setting the inner function equal to

2

gives

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

. Multiplying both sides by

x-1

results in

x+3 = 2(x-1) \implies x+3 = 2x-2 \implies x = 5

.

4

Solve the equation for the second restricted value of the outer function

x = -\frac{1}{3}

Setting the inner function equal to

-2

gives

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

. Multiplying both sides by

x-1

results in

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

.

5

Combine all restricted values of

x

S = \left\{-\frac{1}{3}, 1, 5\right\}

The values of

x

for which the composite function is undefined are

x = 1

(from the inner function's domain restriction) along with

x = 5

and

x = -\frac{1}{3}

(which make the inner function's output equal to the outer function's restricted inputs).

Key Concept

Domain of Composite Functions

Estimated Time:2m 30s

Question 23Question →

Two functions, $f$ and $g$ , are defined as $f(x) = (x - 4)^2$ and $g(x) = |2x - 3|$ . What is the value of the composite function $f(g(x))$ evaluated at $x = -1$ ?

1

9

47

81

125

Show answer & explanation

Answer: 1

Answer

1

To find

f(g(-1))

, we first evaluate the inner function

g

at

x = -1

. Substituting

-1

into

g(x) = |2x - 3|

gives

g(-1) = |2(-1) - 3| = |-5| = 5

. Next, we use this result as the input for the outer function

f

, evaluating

f(5)

. Substituting

5

into

f(x) = (x - 4)^2

gives

f(5) = (5 - 4)^2 = 1^2 = 1

. This corresponds to the option with value 1.

Step-by-Step Solution

1

Evaluate the inner function

g(-1)

first.

g(-1) = |2(-1) - 3| = |-2 - 3| = |-5| = 5

The input to the outer function of a composition is the output of the inner function.

2

Evaluate the outer function

f(x)

at the result from the previous step.

f(5) = (5 - 4)^2 = 1^2 = 1

We substitute the value of

g(-1)

, which is

5

, into the function

f(x)

to find the final value of the composition.

Key Concept

Evaluating a composite function involves applying the inner function first, then using that output as the input for the outer function.

Question 24Question →

Consider the functions $f(x) = \frac{x+3}{x-1}$ , where $x \neq 1$ , and $g(x) = x^2 - x - 4$ . If $x$ is an integer such that the composite function evaluation $g(f(g(x))) = 16$ , what is the product of all such integer values of $x$ ?

Show answer & explanation

Answer: -6

Answer

The product of all integer values of

x

that satisfy the equation is

-6

.

Solving the nested composite equation

g(f(g(x))) = 16

systematically yields the integer solutions

x = 3

and

x = -2

. The product of these values is

-6

.

Step-by-Step Solution

1

Set

u = f(g(x))

and solve the outer quadratic equation

g(u) = 16

.

u = 5

or

u = -4

This simplifies the nested composition into a single-variable quadratic equation.

2

Set

v = g(x)

and solve the rational equations

f(v) = 5

and

f(v) = -4

.

v = 2

or

v = 0.2

This determines the required outputs of the function

g(x)

that will satisfy the composite equation.

3

Solve the quadratic equations

g(x) = 2

and

g(x) = 0.2

for

x

.

x = 3

,

x = -2

, or

x = \frac{5 \pm \sqrt{445}}{10}

This finds all real values of

x

that satisfy the composite equation.

4

Identify the integer values from the solution set and compute their product.

3 \times (-2) = -6

The question specifically requests the product of the integer values of

x

.

Key Concept

Function composition and multi-step equation solving

Question 25Question →

The table below shows some values of the functions $f$ and $g$ for several integer values of $x$ .

$x$	$f(x)$	$g(x)$
$-2$	$3$	$1$
$-1$	$4$	$-2$
$0$	$-1$	$3$
$1$	$2$	$0$
$2$	$-2$	$-1$
$3$	$1$	$2$

What is the value of $f(f(3)) + g(g(-1))$ ?

Show answer & explanation

Answer: 3

Answer

The value of the expression is

3

.

To evaluate

f(f(3)) + g(g(-1))

, we start by evaluating the innermost expressions. Looking at the table for

x = 3

, we find

f(3) = 1

. Looking at the table for

x = -1

, we find

g(-1) = -2

. Substituting these values into the outer functions gives

f(1) + g(-2)

. Using the table again, we look up

x = 1

to find

f(1) = 2

, and look up

x = -2

to find

g(-2) = 1

. Adding these two results yields

2 + 1 = 3

.

Step-by-Step Solution

1

Find the inner function values from the given table.

f(3) = 1

and

g(-1) = -2

Before evaluating the composite functions, we must find the outputs of the innermost functions.

2

Evaluate the outer functions using the results from the first step.

f(f(3)) = f(1) = 2

and

g(g(-1)) = g(-2) = 1

Substitute the inner outputs into the outer functions and look up the new inputs in the table.

3

Add the two resulting values together.

2 + 1 = 3

Combine the two terms to find the total sum requested by the question.

Key Concept

Evaluating composite functions using tables of values

Estimated Time:1m 0s

Question 26Question →

Let the functions $f$ and $g$ be defined for all real numbers by $f(x) = |2x - 6| - 16$ and $g(x) = (x - 3)^2 - 5$ . What is the sum of all real values of $x$ that satisfy the equation $f(g(x)) = 0$ ?

0

6

9

10

12

Show answer & explanation

Answer: 9

Answer

9

The correct answer is 9. First, substitute the inner function into the outer function to get

f(g(x)) = |2g(x) - 6| - 16 = 0

. This simplifies to the absolute value equation

|2g(x) - 6| = 16

. Splitting this into its two possible cases gives

2g(x) - 6 = 16 \implies g(x) = 11

, and

2g(x) - 6 = -16 \implies g(x) = -5

. Solving the first case,

(x-3)^2 - 5 = 11 \implies (x-3)^2 = 16

. Taking both square roots yields

x - 3 = 4 \implies x = 7

and

x - 3 = -4 \implies x = -1

. Solving the second case,

(x-3)^2 - 5 = -5 \implies (x-3)^2 = 0 \implies x - 3 = 0 \implies x = 3

. The sum of all unique real values of

x

that satisfy the original equation is

7 + (-1) + 3 = 9

.

Step-by-Step Solution

1

Set up the composite function equation using the outer function definition.

|2g(x) - 6| - 16 = 0

Substitute the expression of

g(x)

as the input variable into

f(x)

.

2

Isolate the absolute value term and split the equation into two cases.

|2g(x) - 6| = 16 \implies 2g(x) - 6 = 16

or

2g(x) - 6 = -16

An absolute value equation of the form

|A| = B

where

B > 0

has two solutions:

A = B

and

A = -B

.

3

Solve for the target values of

g(x)

in each case.

g(x) = 11

and

g(x) = -5

Isolate the function

g(x)

by adding 6 and dividing by 2 on both sides of each equation.

4

Solve the first case

g(x) = 11

for the variable

x

.

(x-3)^2 - 5 = 11 \implies (x-3)^2 = 16 \implies x - 3 = \pm 4

, yielding

x = 7

and

x = -1

Substitute the algebraic rule for

g(x)

, isolate the squared expression, and extract the square roots.

5

Solve the second case

g(x) = -5

for the variable

x

.

(x-3)^2 - 5 = -5 \implies (x-3)^2 = 0 \implies x - 3 = 0

, yielding

x = 3

Substitute the algebraic rule for

g(x)

, isolate the squared expression, and solve for

x

.

6

Sum all unique real solutions found.

7 + (-1) + 3 = 9

Add the distinct solutions

x = 7

,

x = -1

, and

x = 3

to find the total sum.

Key Concept

Evaluating and solving equations involving composite functions, absolute values, and quadratic expressions.

Question 27Question →

Let the functions $f$ and $g$ be defined for all real numbers by $f(x) = 2x - 3$ and $g(x) = x^2 - 5$ . If $g(f(x)) = 11$ and $x < 0$ , what is the value of $x$ ?

\frac{7}{2}

-\sqrt{12}

-\frac{\sqrt{7}}{2}

-\frac{1}{2}

-\frac{7}{2}

Show answer & explanation

Answer: $-\frac{1}{2}$

Answer

-\frac{1}{2}

The correct answer is

-\frac{1}{2}

. First, the composite function

g(f(x))

is found by substituting

f(x) = 2x - 3

into

g(x) = x^2 - 5

, giving

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

. Setting this expression equal to

11

yields the equation

(2x - 3)^2 - 5 = 11

. Adding

5

to both sides results in

(2x - 3)^2 = 16

. Taking the square root of both sides gives

2x - 3 = \pm 4

. Since the problem specifies that

x < 0

, the expression

2x - 3

must be negative because

2x < 0

and subtracting

3

makes the result less than

-3

. Thus, we set

2x - 3 = -4

. Adding

3

to both sides yields

2x = -1

, and dividing by

2

gives

x = -\frac{1}{2}

.

Step-by-Step Solution

1

Substitute the expression for

f(x)

into the function

g(x)

to obtain the composite function

g(f(x))

.

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

To find

g(f(x))

, we substitute the entire function

f(x)

in place of the input variable in

g(x)

.

2

Set the composite function equal to

11

and isolate the squared term.

(2x - 3)^2 - 5 = 11 \Rightarrow (2x - 3)^2 = 16

We are given that

g(f(x)) = 11

, and adding

5

to both sides isolates the squared binomial.

3

Solve for

2x - 3

by taking the square root of both sides, applying the constraint

x < 0

.

2x - 3 = -4

Since

x < 0

, it follows that

2x < 0

, which means

2x - 3 < -3

. Because

2x-3

must be negative, we take the negative square root of

16

.

4

Solve the linear equation

2x - 3 = -4

for

x

.

2x = -1 \Rightarrow x = -\frac{1}{2}

Adding

3

to both sides gives

2x = -1

, and dividing by

2

yields the final value of

x

.

Key Concept

Function Composition and Solving Quadratic/Linear Equations

Estimated Time:1m 30s

Question 28Question →

Let the functions $f$ and $g$ be defined by $f(x) = (x - 3)^2$ and $g(x) = 2x + 1$ . What is the value of the composite function $f(g(-2))$ ?

Show answer & explanation

Answer: 36

Answer

36

To evaluate the composite function

f(g(-2))

, evaluate from the inside out. First, evaluate the inner function

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

. Next, substitute this output value into the outer function

f(x)

to get

f(-3) = (-3 - 3)^2 = (-6)^2 = 36

.

Step-by-Step Solution

1

Evaluate the inner function

g(x)

at

x = -2

g(-2) = -3

Before evaluating the outer function

f

, we must determine the output of the inner function

g

at the given input value.

2

Evaluate the outer function

f(x)

at the result of the inner function

f(-3) = 36

Substitute the inner output

-3

as the input for

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

, giving

(-3 - 3)^2 = (-6)^2 = 36

.

Key Concept

Function composition and evaluation

Estimated Time:1m 0s

Question 29Question →

For the functions $f(x) = (x - 3)^2 - 2$ and $g(x) = |3x - 10|$ , what is the value of the composite function $g(f(1))$ ?

-4

4

24

28

40

Show answer & explanation

Answer: 4

Answer

The correct answer is 4

Evaluating

f(1)

gives

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

. Substituting this result into

g(x)

gives

g(2) = |3(2) - 10| = |6 - 10| = |-4| = 4

. Therefore, the value of the composite function is 4.

Step-by-Step Solution

1

Evaluate the inner function

f(x)

at

x = 1

.

f(1) = (1 - 3)^2 - 2 = (-2)^2 - 2 = 4 - 2 = 2

To evaluate the composite function

g(f(1))

, the output of the inner function

f(1)

must be calculated first.

2

Substitute the output

f(1) = 2

into the outer function

g(x)

.

g(2) = |3(2) - 10| = |6 - 10| = |-4| = 4

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

g(x)

.

Key Concept

Evaluating a composite function by first calculating the value of the inner function and then using that result as the input for the outer function.

Estimated Time:1m 0s

Question 30Question →

For all real numbers $x$ such that $x \neq 1$ , the function $f$ is defined by $f(x) = \frac{12}{x - 1}$ . For all real numbers $x$ , the function $g$ is defined by $g(x) = x^2 + 2$ . What is the value of the composite function $f(g(3))$ ?

Show answer & explanation

Answer: 1.2

Answer

1.2

To evaluate the composite function

f(g(3))

, we work from the inside out. First, evaluate the inner function

g(3) = 3^2 + 2 = 11

. Next, substitute this result into the outer function

f(x)

to get

f(11) = \frac{12}{11 - 1} = \frac{12}{10} = 1.2

.

Step-by-Step Solution

1

Evaluate the inner function

g(x)

at

x = 3

.

g(3) = 11

To evaluate a composite function of the form

f(g(x))

at a given value, we must first find the output of the inner function,

g(x)

, at that value.

2

Substitute the output from Step 1 as the input for the outer function

f(x)

and evaluate.

f(g(3)) = 1.2

Using the result

g(3) = 11

as the input for

f(x)

gives

f(11) = \frac{12}{11 - 1} = 1.2

.

Key Concept

Function Composition and Evaluation

Question 31Question →

Two functions are defined as $f(x) = (x - 3)^2 - 16$ and $g(x) = |x - 1| - 8$ . If $g(f(x)) = 0$ , what is the sum of all positive values of $x$ ?

4

8

10

14

24

Show answer & explanation

Answer: $14$

Answer

The correct sum of all positive values of

x

is 14.

To find the sum of all positive values of

x

that satisfy

g(f(x)) = 0

, first substitute

f(x)

into

g(x)

to get the equation

|f(x) - 1| - 8 = 0

. This simplifies to

|f(x) - 1| = 8

. Solving this absolute value equation yields two cases:

f(x) - 1 = 8

or

f(x) - 1 = -8

.

For the first case,

f(x) = 9

. Substituting

f(x) = (x - 3)^2 - 16

gives

(x - 3)^2 - 16 = 9 \Rightarrow (x - 3)^2 = 25

. Taking the square root of both sides gives

x - 3 = 5

or

x - 3 = -5

, which results in

x = 8

or

x = -2

. The only positive solution from this case is

8

.

For the second case,

f(x) = -7

. Substituting

f(x) = (x - 3)^2 - 16

gives

(x - 3)^2 - 16 = -7 \Rightarrow (x - 3)^2 = 9

. Taking the square root of both sides gives

x - 3 = 3

or

x - 3 = -3

, which results in

x = 6

or

x = 0

. The only positive solution from this case is

6

(since

0

is not positive).

Adding the positive solutions gives

8 + 6 = 14

.

Step-by-Step Solution

1

Set up the composite equation

g(f(x)) = 0

.

|f(x) - 1| - 8 = 0 \Rightarrow |f(x) - 1| = 8

Substitute the expression for

f(x)

into

g(x)

to establish the relationship.

2

Solve the absolute value equation by separating it into two distinct cases.

f(x) - 1 = 8 \Rightarrow f(x) = 9

or

f(x) - 1 = -8 \Rightarrow f(x) = -7

An absolute value equation

|u| = c

splits into

u = c

and

u = -c

when

c \geq 0

.

3

Substitute the definition of

f(x) = (x - 3)^2 - 16

into the first case and solve for

x

.

(x - 3)^2 - 16 = 9 \Rightarrow (x - 3)^2 = 25 \Rightarrow x - 3 = \pm 5 \Rightarrow x = 8

or

x = -2

Isolate the squared binomial and take the square root of both sides to find all real solutions for this case.

4

Substitute the definition of

f(x) = (x - 3)^2 - 16

into the second case and solve for

x

.

(x - 3)^2 - 16 = -7 \Rightarrow (x - 3)^2 = 9 \Rightarrow x - 3 = \pm 3 \Rightarrow x = 6

or

x = 0

Isolate the squared binomial and take the square root of both sides to find all real solutions for this case.

5

Identify the positive solutions and calculate their sum.

The positive solutions are

8

and

6

. The sum is

8 + 6 = 14

.

Exclude non-positive values (

-2

is negative, and

0

is neither positive nor negative) and add the remaining values.

Key Concept

Function composition involves substituting one function into another, and evaluating the resulting composite equation requires solving multi-step equations including absolute value and quadratic relations.

Question 32Question →

Given the functions $f(x) = \sqrt{3x + 4}$ and $g(x) = x^2 - 5$ , what is the value of $g(f(7))$ ?

Show answer & explanation

Answer: 20

Answer

20

Evaluating the inner function first gives

f(7) = \sqrt{3(7) + 4} = 5

. Substituting this value into the outer function yields

g(5) = 5^2 - 5 = 20

.

Step-by-Step Solution

1

Evaluate the inner function

f(x)

at

x = 7

.

f(7) = 5

To evaluate a composite function of the form

g(f(x))

, we first calculate the value of the inner function

f(x)

at the given input.

2

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

g(x)

.

g(5) = 20

Since

f(7) = 5

, evaluating

g(f(7))

is equivalent to evaluating

g(5)

.

Key Concept

Function composition involves evaluating an inner function and then using that result as the input for an outer function.

Question 33Question →

The functions $f$ and $g$ are defined for the integers $-2 \leq x \leq 3$ by the table below.

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

What is the value of $x$ for which $g(f(x)) = -1$ ?

-2

-1

0

1

3

Show answer & explanation

Answer: $1$

Answer

The value of

x

is

1

.

To solve

g(f(x)) = -1

, we work from the outside in. First, we find the input to the outer function

g

that results in an output of

-1

. Looking at the table,

g(0) = -1

, which means the inner expression

f(x)

must equal

0

. Next, we find the input to

f

that results in an output of

0

. Looking at the table,

f(1) = 0

. Therefore, the value of

x

that satisfies the equation is

1

.

Step-by-Step Solution

1

Identify the outer function in the composition

g(f(x)) = -1

.

The outer function is

g

, so we must first find an input

u

such that

g(u) = -1

.

This allows us to break down the composition into two simpler function evaluation steps.

2

Look at the column for

g(x)

in the table to find where the output is

-1

.

The table shows

g(0) = -1

, meaning the input to

g

must be

0

. Therefore,

f(x) = 0

.

This establishes the required output value for the inner function

f(x)

.

3

Look at the column for

f(x)

in the table to find where the output is

0

.

The table shows

f(1) = 0

, meaning the input

x

must be

1

.

This provides the final value of the variable

x

that satisfies the original equation.

Key Concept

Evaluating composite functions using a table of values

Estimated Time:1m 0s

Question 34Question →

For the functions $f(x) = |4x - 17|$ and $g(x) = 3 - 2x$ , what is the value of $f(g(5))$ ?

Show answer & explanation

Answer: 45

Answer

The correct answer is 45.

Evaluating the inner function first yields

g(5) = 3 - 2(5) = -7

. Substituting this result into the outer function gives

f(-7) = |4(-7) - 17| = |-45| = 45

.

Step-by-Step Solution

1

Evaluate the inner function

g(5)

g(5) = 3 - 2(5) = -7

In a composite function of the form

f(g(x))

, the inner function

g(x)

must be evaluated first at the given input.

2

Evaluate the outer function

f(x)

at the output of the inner function

f(-7) = |4(-7) - 17| = |-28 - 17| = |-45| = 45

The output of the inner function,

-7

, becomes the input for the outer function

f(x)

.

Key Concept

Function Evaluation and Composition

Question 35Question →

For the functions $f(x) = (x - 2)^2 - 5$ and $g(x) = |x + 1| - 3$ , what is the value of $f(g(-2))$ ?

9

-5

11

31

-22

Show answer & explanation

Answer: 11

Answer

11

To find the value of

f(g(-2))

, first evaluate the inner function

g(x)

at

x = -2

. Substituting

-2

into

g(x) = |x + 1| - 3

gives

g(-2) = |-2 + 1| - 3 = 1 - 3 = -2

. Next, substitute this output as the input for the outer function

f(x)

. Substituting

-2

into

f(x) = (x - 2)^2 - 5

yields

f(-2) = (-2 - 2)^2 - 5 = (-4)^2 - 5 = 16 - 5 = 11

.

Step-by-Step Solution

1

Evaluate the inner function at the given input value

g(-2) = |-2 + 1| - 3 = |-1| - 3 = 1 - 3 = -2

In function composition

f(g(x))

, the inner function must be evaluated first to determine the input for the outer function.

2

Evaluate the outer function using the output of the inner function

f(-2) = (-2 - 2)^2 - 5 = (-4)^2 - 5 = 16 - 5 = 11

Substitute the result from the inner function evaluation into the outer function to find the final value.

Key Concept

Function composition and evaluation

Estimated Time:1m 0s

Question 36Question →

A temperature control system in a chemical reactor regulates target output temperature, in degrees Celsius, based on the reactor's pressure coefficient. The pressure coefficient $P$ is determined by the elapsed time $t$ , in minutes, according to the function $P(t) = (t - 3)^2 - 5$ . The target temperature $T$ is determined by the pressure coefficient $P$ according to the function $T(P) = 3|P - 2| + 1$ . What is the target output temperature of the system, in degrees Celsius, at time $t = 1$ minute?

-8

12

10

-4

46

Show answer & explanation

Answer: 10

Answer

10

The correct temperature is 10. By first finding the pressure at

t = 1

to be

P(1) = (1 - 3)^2 - 5 = 4 - 5 = -1

, and then substituting this pressure into the temperature function, we get

T(-1) = 3|-1 - 2| + 1 = 3(3) + 1 = 10

.

Step-by-Step Solution

1

Calculate the pressure coefficient

P(1)

at time

t = 1

minute by substituting

t = 1

into

P(t) = (t - 3)^2 - 5

.

P(1) = -1

Before calculating the temperature, we must find the input value

P

corresponding to

t = 1

.

2

Evaluate the target temperature

T(P)

at the pressure coefficient

P = -1

by substituting

P = -1

into

T(P) = 3|P - 2| + 1

.

T(-1) = 10

Applying the absolute value definition

|-3| = 3

, we find

3(3) + 1 = 10

.

Key Concept

Evaluating composite functions in a real-world scenario.

Alternative Method

Instead of evaluating step-by-step numerically, we can write the composite function symbolically:

T(P(t)) = 3|((t - 3)^2 - 5) - 2| + 1 = 3|(t - 3)^2 - 7| + 1

. Plugging in

t = 1

yields

T(P(1)) = 3|(1 - 3)^2 - 7| + 1 = 3|4 - 7| + 1 = 3|-3| + 1 = 10

.

Estimated Time:1m 30s

Question 37Question →

The functions $f$ and $g$ are defined for all permissible real numbers by $f(x) = \frac{x + 3}{x - 1}$ and $g(x) = 2x - 5$ . If $(f \circ g)(x) = 3$ , what is the value of $x$ ?

Show answer & explanation

Answer: 4

Answer

The value of

x

that satisfies the equation

(f \circ g)(x) = 3

is

4

.

To solve for

x

in

(f \circ g)(x) = 3

, we find the composite function

f(g(x))

by substituting

g(x) = 2x - 5

into

f(x)

. This yields

f(g(x)) = \frac{(2x-5)+3}{(2x-5)-1} = \frac{2x-2}{2x-6}

. Setting this equal to

3

gives

\frac{2x-2}{2x-6} = 3

. Multiplying by

2x-6

yields

2x-2 = 6x-18

. Rearranging terms to isolate

x

gives

4x = 16

, which results in

x = 4

.

Step-by-Step Solution

1

Substitute the expression for

g(x)

into

f(x)

to obtain the composite function

(f \circ g)(x)

.

(f \circ g)(x) = \frac{2x - 2}{2x - 6}

By definition of function composition,

(f \circ g)(x) = f(g(x))

. Substituting

g(x) = 2x - 5

into

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

yields

f(2x-5) = \frac{(2x-5)+3}{(2x-5)-1}

, which simplifies to

\frac{2x-2}{2x-6}

.

2

Set the composite function expression equal to

3

and solve for

x

.

x = 4

We set

\frac{2x-2}{2x-6} = 3

. Multiplying both sides by the denominator

2x-6

gives

2x-2 = 3(2x-6)

. Expanding the right side gives

2x-2 = 6x-18

. Subtracting

2x

from both sides and adding

18

to both sides results in

16 = 4x

. Dividing by

4

gives

x = 4

.

Key Concept

Function Composition and Evaluation

Estimated Time:1m 30s

Question 38Question →

Let the functions $f$ and $g$ be defined by $f(x) = |x - 4|$ and $g(x) = 2x + 1$ for all real numbers. For what values of $x$ does the composition $f(g(x)) = 5$ ?

x = -1

and

x = 4

x = 4

only

x = 2

and

x = 6

x = -2

and

x = 2

x = -4

and

x = 4

Show answer & explanation

Answer: $x = -1$ and $x = 4$

Answer

x = -1

and

x = 4

To find the values of

x

for which

f(g(x)) = 5

, we substitute the expression for

g(x)

into

f(x)

, yielding

|(2x + 1) - 4| = 5

, which simplifies to

|2x - 3| = 5

. Setting up the two possible cases for the absolute value gives

2x - 3 = 5

(which solves to

x = 4

) and

2x - 3 = -5

(which solves to

x = -1

). Both values of

x

satisfy the original composition equation.

Step-by-Step Solution

1

Substitute the inner function

g(x) = 2x + 1

into the outer function

f(x) = |x - 4|

to find the composite function

f(g(x))

.

f(g(x)) = |(2x + 1) - 4| = |2x - 3|

Function composition requires evaluating the outer function at the output of the inner function.

2

Set the composite expression equal to

5

.

|2x - 3| = 5

We are solving for the values of

x

where the output of the composite function is

5

.

3

Split the absolute value equation into its positive and negative cases and solve each linear equation for

x

.

Case 1:

2x - 3 = 5 \implies 2x = 8 \implies x = 4

. Case 2:

2x - 3 = -5 \implies 2x = -2 \implies x = -1

.

An absolute value equation

|u| = c

(with

c > 0

) is equivalent to

u = c

or

u = -c

.

Key Concept

Evaluating a composite function with an absolute value and solving the resulting equations by considering both positive and negative cases.

Estimated Time:1m 30s

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