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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2 marks

The function space f is decreasing and is defined for all real numbers. The table gives values for space f open parentheses x close parentheses at selected values of x.

space x

-2

-1

0

1

2

space f open parentheses x close parentheses

14

7

3.5

1.75

0.875

The function g is given by g open parentheses x close parentheses equals negative 0.167 x cubed plus x squared minus 1.834.

(i) The function h is defined by h open parentheses x close parentheses equals open parentheses g ring operator f close parentheses open parentheses x close parentheses equals g open parentheses f open parentheses x close parentheses close parentheses. Find the value of h open parentheses 1 close parentheses as a decimal approximation, or indicate that it is not defined. Show the work that leads to your answer.

(ii) Find the value of space f to the power of negative 1 end exponent open parentheses 3.5 close parentheses, or indicate that it is not defined.

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2a
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2 marks
Graph of a function on a grid, with a curve starting at x=-3.5 and ending at x=3.5. From the start, it dips down, then turns back up, and then dips down again before getting to x=3.5. The points (-3, 1), (0, 1) and (3, 1) are on the curve.

The figure shows the graph of the function space f on its domain of negative 3.5 less or equal than x less or equal than 3.5. The points left parenthesis negative 3 comma space 1 right parenthesis, left parenthesis 0 comma space 1 right parenthesis, and left parenthesis 3 comma space 1 right parenthesis are on the graph of space f. The function g is given by g open parentheses x close parentheses equals 2.916 times open parentheses 0.7 close parentheses to the power of x.

(i) The function h is defined by h open parentheses x close parentheses equals open parentheses g ring operator f close parentheses open parentheses x close parentheses equals g open parentheses f open parentheses x close parentheses close parentheses. Find the value of h open parentheses 3 close parentheses as a decimal approximation, or indicate that it is not defined.

(ii) Find all values of x for which space f open parentheses x close parentheses equals 1, or indicate that there are no such values.

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2b
2 marks

(i) Determine if space f has an inverse function.

(ii) Give a reason for your answer based on the definition of a function and the graph of y equals f open parentheses x close parentheses.

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3
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1 mark

The function h is given by h open parentheses x close parentheses equals open parentheses 2 x minus 5 close parentheses cubed.

Express h open parentheses x close parentheses as a composition space f open parentheses g open parentheses x close parentheses close parentheses, where space f and g are functions and neither space f nor g is the identity function. Give explicit expressions for space f open parentheses x close parentheses and g open parentheses x close parentheses.

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4
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1 mark

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

Find the exact value of space f open parentheses f to the power of negative 1 end exponent open parentheses 11 close parentheses close parentheses.

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5
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1 mark

The function space f is increasing and is defined for 0 less or equal than x less or equal than 5. The range of space f is - 2 \leq y \leq 18. Let space f to the power of negative 1 end exponent be the inverse function of space f.

State the domain and range of space f to the power of negative 1 end exponent.

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6
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2 marks

The graph of a function space f in the x y-plane passes through the points open parentheses negative 1 comma 0 close parentheses and open parentheses 2 comma 5 close parentheses. Let space f to the power of negative 1 end exponent be the inverse function of space f.

State the coordinates of two points that must lie on the graph of space f to the power of negative 1 end exponent. Then describe how the graph of space f to the power of negative 1 end exponent relates geometrically to the graph of space f.

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7
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2 marks

The function space f is increasing and is defined for x greater than 0. The table gives values of space f open parentheses x close parentheses at selected values of x.

x

1

2

4

8

16

space f open parentheses x close parentheses

3

5

7

9

11

The function g is given by g open parentheses x close parentheses equals negative 0.1371 x to the power of 4 plus x cubed minus 1.2 x squared.

(i) The function h is defined by h open parentheses x close parentheses equals open parentheses g ring operator f close parentheses open parentheses x close parentheses equals g open parentheses f open parentheses x close parentheses close parentheses. Find the value of h open parentheses 2 close parentheses as a decimal approximation, or indicate that it is not defined. Show the work that leads to your answer.

(ii) Find the value of space f to the power of negative 1 end exponent open parentheses 7 close parentheses, or indicate that it is not defined.

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8
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2 marks

The function space f is increasing and is defined for x greater or equal than 0. The table gives values of space f open parentheses x close parentheses at selected values of x.

x

0

1

2

3

4

space f open parentheses x close parentheses

3

5

9

15

23

The function g is given by g open parentheses x close parentheses equals fraction numerator 2 x cubed minus 5 x plus 1 over denominator x minus 3 end fraction.

(i) The function h is defined by h open parentheses x close parentheses equals open parentheses g ring operator f close parentheses open parentheses x close parentheses equals g open parentheses f open parentheses x close parentheses close parentheses. Find the value of h open parentheses 2 close parentheses as a decimal approximation, or indicate that it is not defined. Show the work that leads to your answer.

(ii) Find the value of space f to the power of negative 1 end exponent open parentheses 5 close parentheses, or indicate that it is not defined.

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9
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2 marks

The function space f is given by space f open parentheses x close parentheses equals fraction numerator 3 x plus 1 over denominator x minus 2 end fraction for space x not equal to 2.

Find an expression for space f to the power of negative 1 end exponent open parentheses x close parentheses. Show the work that leads to your answer.

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10
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2 marks

The number of mugs a small company sells per week depends on the amount spent on advertising, x dollars, according to the function N given by N open parentheses x close parentheses equals 30 plus 0.05 x.

The weekly revenue, in dollars, depends on the number of mugs sold, N, according to the function R given by R open parentheses N close parentheses equals 7.50 times N minus 80.

Find the value of open parentheses R ring operator N close parentheses open parentheses 400 close parentheses. Interpret the meaning of your answer in the context of the problem.

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11
2 marks

The function space f is given by space f open parentheses x close parentheses equals square root of x minus 5 end root for x greater or equal than 5. The function g is given by g open parentheses x close parentheses equals x squared plus 1 for all real numbers x.

(i) Find a simplified expression for open parentheses f ring operator g close parentheses open parentheses x close parentheses equals f open parentheses g open parentheses x close parentheses close parentheses.

(ii) State the domain of f ring operator g.

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12
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2 marks

The function space f is given by space f open parentheses x close parentheses equals fraction numerator 2 x plus 3 over denominator x minus 1 end fraction for x not equal to 1. The function g is given by g open parentheses x close parentheses equals x squared plus 4 for x greater or equal than 0.

(i) Find an expression for space f to the power of negative 1 end exponent open parentheses x close parentheses.

(ii) Find the value of space f to the power of negative 1 end exponent open parentheses g open parentheses 2 close parentheses close parentheses as a decimal approximation. Show the work that leads to your answer.

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13
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2 marks

The function space f is an increasing function defined for all real numbers. The table gives values of space f at selected values of x.

x

- 2

0

1

3

5

space f open parentheses x close parentheses

2

6

8

10

12

The function g is given by g open parentheses x close parentheses equals x squared minus 2 x plus 5 for x greater or equal than 1.

(i) Find the value of space f to the power of negative 1 end exponent open parentheses 2 close parentheses.

(ii) Find the value of g to the power of negative 1 end exponent open parentheses f open parentheses 0 close parentheses close parentheses as a decimal approximation. Show the work that leads to your answer.

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