AP®MathsCollege BoardPrecalculusExam QuestionsExponential & Logarithmic FunctionsComposite & Inverse Functions

Composite & Inverse Functions (College Board AP® Precalculus): Exam Questions

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Composite & Inverse Functions

1
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1 mark

The function h is given by h open parentheses x close parentheses equals open parentheses 3 x minus 1 close parentheses squared. If h open parentheses x close parentheses equals f open parentheses g open parentheses x close parentheses close parentheses and space f open parentheses x close parentheses equals x squared, which of the following is an expression for g open parentheses x close parentheses?

  • g open parentheses x close parentheses equals 3 x

  • g open parentheses x close parentheses equals 3 x minus 1

  • g open parentheses x close parentheses equals square root of 3 x minus 1 end root

  • g open parentheses x close parentheses equals open parentheses 3 x minus 1 close parentheses squared

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2
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1 mark

The function space f is given by space f open parentheses x close parentheses equals 2 x plus 1. Let space f to the power of negative 1 end exponent be the inverse function of space f. What is the value of space f open parentheses f to the power of negative 1 end exponent open parentheses 7 close parentheses close parentheses?

  • 3

  • 7

  • 15

  • 31

Choose your answer

3
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1 mark

The function space f has domain open square brackets 2 comma 8 close square brackets and range open square brackets negative 3 comma 5 close square brackets. Let space f to the power of negative 1 end exponent be the inverse function of space f. What is the domain of space f to the power of negative 1 end exponent?

  • open square brackets negative 8 comma negative 2 close square brackets

  • open square brackets negative 5 comma 3 close square brackets

  • open square brackets negative 3 comma 5 close square brackets

  • open square brackets 2 comma 8 close square brackets

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4
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1 mark

The function f is given by f \left(x\right) = x^{2} - 2, and the function g is given by g \left(x\right) = \frac{x + 1}{x}. Which of the following is an expression for f \left(g \left(x\right)\right) ?

  • fraction numerator x squared minus 1 over denominator x squared minus 2 end fraction

  • \frac{- x^{2} + 2 x + 1}{x^{2}}

  • \frac{- x^{2} + 1}{x^{2}}

  • \frac{x^{2} + 2 x - 1}{x^{2}}

Choose your answer

5
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1 mark

The table gives values for the functions f and g at selected values of x. Functions f and g are defined for all real numbers. Let h be the function defined by h \left(x\right) = f \left(g \left(x\right)\right). What is the value of h \left(1\right) ?

x

- 2

- 1

0

1

2

f \left(x\right)

3

- 1

2

0

- 2

g \left(x\right)

1

2

- 1

- 2

0

  • 3

  • 0

  • - 2

  • - 1

Choose your answer

6
1 mark

The graph of the piecewise-linear function f is shown in the figure. Let g be the inverse function of f. What is the maximum value of g ?

Graph of a piecewise-linear function f consisting of two line segments: from (2, 3) to (5, 6), and from (5, 6) to (8, 14).

  • \frac{1}{14}

  • \frac{1}{8}

  • 8

  • 14

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7
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1 mark

The function h is given by h left parenthesis x right parenthesis equals cos open parentheses x over 2 close parentheses minus sin open parentheses x over 2 close parentheses and has a period of 4\pi. In order to define the inverse function of h, which of the following specifies a restricted domain for h and provides a rationale for why h is invertible on that domain?

  • 0 \leq x \leq 2\pi, because all possible values of h(x) occur without repeating on this interval

  • -\frac{\pi}{2} \leq x \leq \frac{3\pi}{2}, because all possible values of h(x) occur without repeating on this interval

  • 0 \leq x \leq 2\pi, because the length of this interval is half of the period

  • -\frac{\pi}{2} \leq x \leq \frac{3\pi}{2}, because the length of this interval is half of the period

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8
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1 mark

The table gives values of the function g for selected values of x. The function f is given by f(x) = 2^x + x^2. What is the value of f(g(3))?

x

-3

0

3

5

36

g(x)

5

\frac{1}{2}

-2

4

9

  • -\frac{7}{4}

  • \frac{17}{4}

  • 17

  • 32

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9
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1 mark

The function f is given by f\left(x\right) = x^{2} + 1, and the function g is given by g\left(x\right) = \frac{x - 1}{x}. Which of the following is an expression for f\left(g\left(x\right)\right)?

  • \frac{x^{2}}{x^{2} + 1}

  • \frac{2x^{2} - 2x + 1}{x^{2}}

  • \frac{2x^{2} + 1}{x^{2}}

  • \frac{x^{2} - 2x + 2}{x^{2}}

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10
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1 mark

A factory produces widgets. The cost per widget, in dollars, depends on the number of widgets produced, q, according to the function P given by P open parentheses q close parentheses equals 12 minus 0.03 q. The number of widgets produced each day depends on the day's temperature T, in degrees Celsius, according to the function q given by q open parentheses T close parentheses equals 80 plus 1.5 T.

When the temperature is T equals 22 degrees Celsius, what is the cost per widget, in dollars?

  • $3.39

  • $8.61

  • $11.34

  • $15.39

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11
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1 mark

The functions space f and g are given by space f open parentheses x close parentheses equals square root of x minus 1 end root and g open parentheses x close parentheses equals x squared. Which of the following is the domain of the composite function space f ring operator g?

  • All real numbers

  • - 1 \leq x \leq 1

  • x \geq 1

  • x \leq - 1 or x \geq 1

Choose your answer

12
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1 mark

The function space f is given by space f open parentheses x close parentheses equals fraction numerator 2 x over denominator x minus 1 end fraction for x \neq 1. Which of the following is an expression for space f to the power of negative 1 end exponent open parentheses x close parentheses?

  • fraction numerator 2 x over denominator x minus 1 end fraction

  • fraction numerator x minus 1 over denominator 2 x end fraction

  • fraction numerator x over denominator x minus 2 end fraction

  • fraction numerator x over denominator x plus 2 end fraction

Choose your answer

13
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1 mark

The function space f is given by space f open parentheses x close parentheses equals x squared minus 4 for x \geq 0. Which of the following is an expression for space f to the power of negative 1 end exponent open parentheses x close parentheses?

  • \pm \sqrt{x + 4}

  • \sqrt{x - 4}

  • - \sqrt{x + 4}

  • \sqrt{x + 4}

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14
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1 mark

The functions space f and g are given by space f open parentheses x close parentheses equals square root of 2 x plus 5 end root and g open parentheses x close parentheses equals 3 x minus 1. What is the value of open parentheses f ring operator g close parentheses open parentheses 4 close parentheses?

  • 3 . 606

  • 5 . 196

  • 9 . 817

  • 11 . 000

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15
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1 mark

The function space f is given by space f open parentheses x close parentheses equals x cubed minus 5. What is the value of space f to the power of negative 1 end exponent open parentheses 20 close parentheses?

  • 2 . 466

  • 2 . 714

  • 2 . 924

  • 25 . 000

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16
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1 mark

Let h open parentheses x close parentheses equals square root of x squared plus 4 end root and f open parentheses x close parentheses equals square root of x plus 4 end root. If h open parentheses x close parentheses equals open parentheses f ring operator g close parentheses open parentheses x close parentheses for all real x, which of the following is the function g?

  • g open parentheses x close parentheses equals x squared

  • g open parentheses x close parentheses equals x squared plus 4

  • g open parentheses x close parentheses equals x squared minus 4

  • g open parentheses x close parentheses equals square root of x squared plus 4 end root

Choose your answer

17
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1 mark

The function space f is given by space f open parentheses x close parentheses equals x squared minus 4 x plus 1. In order to define the inverse function space f to the power of negative 1 end exponent, which of the following specifies a restricted domain for space f?

  • - 1 \leq x \leq 5

  • 0 \leq x \leq 4

  • 1 \leq x \leq 3

  • 3 \leq x \leq 7

Choose your answer

18
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1 mark

The function space f is an invertible function such that space f open parentheses 3 close parentheses equals 7 and space f to the power of negative 1 end exponent open parentheses 3 close parentheses equals 5. What is the value of space f open parentheses f open parentheses 5 close parentheses close parentheses?

  • 3

  • 5

  • 7

  • The value cannot be determined from the given information.

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19
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1 mark

The temperature T, in degrees, of a chemical reaction is modeled by the function h given by h open parentheses p close parentheses equals 2 p squared plus 30 for space p greater than 0, where space p is the pressure in atmospheres. The pressure in atmospheres is modeled by the function g given by g open parentheses t close parentheses equals t over 2 plus 1 for t greater or equal than 0, where t is the time in seconds. At what value of t does the temperature reach 70 degrees?

  • 3 . 472

  • 6 . 944

  • 8 . 944

  • 10 . 944

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20
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1 mark

Let space f be the function defined by space f open parentheses x close parentheses equals fraction numerator 3 plus 2 x over denominator 1 minus x end fraction. Which of the following is equal to \left(f \circ f\right) \left(x\right)?

  • \dfrac{3 + 2 x}{1 - x}

  • \dfrac{3 x + 2}{x + 9}

  • \dfrac{x + 9}{3 x + 2}

  • -\dfrac{x + 9}{3 x + 2}

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21
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1 mark

Let space f be the function defined by space f open parentheses x close parentheses equals open parentheses x minus 3 close parentheses squared plus 1 for x greater or equal than 3. What is the value of space f to the power of negative 1 end exponent open parentheses 7 close parentheses?

  • 3 minus square root of 6

  • square root of 6

  • 3 plus square root of 6

  • 3 plus square root of 7

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