Homework 9 (for Test 3)-ariel gonzalez3/16/24, 15:01

Instructor: Shivanni Jagessar

Course: MAC 1105 U02 Jagessar Monday &

Wednesday 11:30 AM

Student: ariel gonzalez

Date: 03/16/24

*1. Write a piecewise function for the given graph.

5

4

3

2

1

-5 -4 -3 -2 -1

-1

Assignment: Homework 9 (for Test 3)

y

x

1 2 3 4 5

-2

-3

-4

-5

What is the rule?

A.

B.

f(x) =

−1

if x ≤ − 2

−3

if x > − 3

f(x) =

C.

−1

if x ≤ − 2

−3

if x > − 2

f(x) =

−1

if x ≤ − 3

−3

if x > − 2

2. Select the answer that best completes the given statement.

The notation f

(1)

−1

means the (1)

of the function f.

inverse

composition

3. Fill in the blanks so that the resulting statement is true.

If the function g is the inverse of the function f, then f(g(x)) = (1)

(1)

x

(2)

y

and g(f(x)) = (2)

x.

y.

4. Select the answers that best complete the given statement.

A function f has an inverse that is a function if there is no (1)

Such a function is called a/an (2)

(1)

horizontal

vertical

(2)

line that intersects the graph of f at more than one point.

function.

one-to-one

inverse

*5. Solve using the test-point method.

2

w −w−6

≥0

w−7

The solution set is

.

(Type your answer in interval notation.)

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Homework 9 (for Test 3)-ariel gonzalez

3/16/24, 15:01

6. Fill in the blank.

The graph of f

The graph of f

(1)

−1

−1

is a reflection of the graph of f about the line whose equation is _______.

is a reflection of the graph of f about the line whose equation is (1)

y = x.

y = 0.

x = 0.

y = − x.

7. Find f(g(x)) and g(f(x)) and determine whether the pair of functions f and g are inverses of each other.

f(x) = 9x and g(x) =

a.

f(g(x)) =

b.

g(f(x)) =

c.

x

9

f and g are inverses of each other.

f and g are not inverses of each other.

8. Watch the video and then solve the problem given below.

Click here to watch the video.1

Find f(g(x)) and g(f(x)) and determine whether the pair of functions f and g are inverses of each other.

f(x) = 2x + 7 and g(x) =

x−7

2

a. f(g(x)) =

(Simplify your answer.)

b. g(f(x)) =

(Simplify your answer.)

c. Are f and g inverses of each other?

Yes

No

1: http://https://mediaplayer.pearsoncmg.com/assets/bzca7e_02_07_01

*9. Let f(x) = 5×2 − 9x + 5. Find f(x + h).

f(x + h) =

(Do not factor.)

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Homework 9 (for Test 3)-ariel gonzalez

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10. Find f(g(x)) and g(f(x)) and determine whether the pair of functions f and g are inverses of each other.

f(x) =

5

5

x − 2 and g(x) = x + 2

f(g(x)) =

g(f(x)) =

Determine whether f and g are inverses of each other. Choose the correct answer below.

A. Functions f and g are inverses of each other.

B. Functions f and g are not inverses of each other.

11. The function f(x) = x + 17 is one-to-one.

a. Find an equation for f

−1

(x), the inverse function.

b. Verify that your equation is correct by showing that f f

−1

(x) = x and f

−1

(f(x)) = x.

a. Select the correct choice below and fill in the answer box(es) to complete your choice.

(Simplify your answer. Use integers or fractions for any numbers in the expression.)

A. f

B. f

C. f

D. f

−1

−1

−1

−1

(x) =

, for all x

(x) =

, for x ≤

(x) =

, for x ≠

(x) =

, for x ≥

b. Verify that the equation is correct.

f f

−1

(x)

= f

and

=

f

−1

(f(x)) = f

=

−1

Substitute.

Simplify.

The equation is (1)

(1)

not verified.

verified.

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Homework 9 (for Test 3)-ariel gonzalez

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12. The function f(x) = 6x is one-to-one.

a. Find an equation for f

−1

(x), the inverse function.

b. Verify that your equation is correct by showing that f f

−1

(x) = x and f

−1

(f(x)) = x.

a. Select the correct choice below and fill in the answer box(es) to complete your choice.

(Simplify your answer. Use integers or fractions for any numbers in the expression.)

A. f

B. f

C. f

D. f

−1

−1

−1

−1

(x) =

, for x ≤

(x) =

, for x ≠

(x) =

for x ≥

(x) =

, for all x

b. Verify that the equation is correct.

f f

−1

(x)

= f

and

=

f

−1

(f(x)) = f

−1

=

Substitute.

Simplify.

(Simplify your answers. Use integers or fractions for any numbers in the expressions.)

The equation is (1)

(1)

verified.

not verified.

*13. Solve the inequality. State the solution in interval notation.

2

a + 18 ≤ 7a

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. The solution set is

. (Type your answer in interval notation.)

B. There is no solution.

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Homework 9 (for Test 3)-ariel gonzalez

3/16/24, 15:01

14. The function f(x) = 5x + 6 is one-to-one.

a. Find an equation for f

−1

, the inverse function.

b. Verify that your equation is correct by showing that f f

−1

(x) = x and f

−1

(f(x)) = x.

a. Select the correct choice below and fill in the answer box(es) to complete your choice.

(Simplify your answer. Use integers or fractions for any numbers in the expression.)

A. f

B. f

C. f

D. f

−1

−1

−1

−1

(x) =

, for x ≤

(x) =

, for x ≥

(x) =

, for x ≠

(x) =

, for all x

b. Verify that the equation is correct.

f f

−1

(x)

= f

and

=

f

−1

(f(x)) = f

−1

Substitute.

=

Simplify.

The equation is (1)

(1)

not verified.

verified.

15. The function f(x) = x3 − 3 is one-to-one.

Find an equation for f

−1

(x)

−1

f (x) =

(Type an expression. Use integers or fractions for any numbers in the expression.)

16. The function f(x) = (x + 6)5 is one-to-one.

Find an equation for f

−1

(x).

−1

f (x) =

(Type an expression. Use integers or fractions for any number in the expression.)

*17.

From the graph of the function, state the domain, the

range, and the intervals on which the function is

increasing, decreasing, or constant. Complete parts

(a) and (b).

(a)

y

(b)

6

4

-4

-2

-2

y

4

2

-6

(a) The domain is

6

2

x

2

4

6

-6

-4

-2

-2

-4

-4

-6

-6

x

2

4

6

.

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Homework 9 (for Test 3)-ariel gonzalez

3/16/24, 15:01

(Type your answer in interval notation.)

The range is

.

(Type your answer in interval notation.)

On what interval(s) is the function increasing? Select the correct choice below and, if necessary, fill in the answer box to complete your

choice.

A. The function is increasing on

.

(Type your answer in interval notation. Use a comma to separate answers as needed.)

B. The function is never increasing.

On what interval(s) is the function decreasing? Select the correct choice below and, if necessary, fill in the answer box to complete your

choice.

A. The function is decreasing on

.

(Type your answer in interval notation. Use a comma to separate answers as needed.)

B. The function is never decreasing.

On what interval(s) is the function constant? Select the correct choice below and, if necessary, fill in the answer box to complete your

choice.

A. The function is constant on

.

(Type your answer in interval notation. Use a comma to separate answers as needed.)

B. The function is never constant.

(b) The domain is

.

(Type your answer in interval notation.)

The range is

.

(Type your answer in interval notation.)

On what interval(s) is the function increasing? Select the correct choice below and, if necessary, fill in the answer box to complete your

choice.

A. The function is increasing on

.

(Type your answer in interval notation. Use a comma to separate answers as needed.)

B. The function is never increasing.

On what interval(s) is the function decreasing? Select the correct choice below and, if necessary, fill in the answer box to complete your

choice.

A. The function is decreasing on

.

(Type your answer in interval notation. Use a comma to separate answers as needed.)

B. The function is never decreasing.

On what interval(s) is the function constant? Select the correct choice below and, if necessary, fill in the answer box to complete your

choice.

A. The function is constant on

.

(Type your answer in interval notation. Use a comma to separate answers as needed.)

B. The function is never constant.

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Page 6 of 19

Homework 9 (for Test 3)-ariel gonzalez

18.

The function f(x) =

3/16/24, 15:01

11

is one-to-one.

x

a. Find an equation for f

−1

(x), the inverse function.

b. Verify that your equation is correct by showing that f f

−1

(x) = x and f

−1

(f(x)) = x.

a. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

(Simplify your answer. Use integers or fractions for any numbers in the expression.)

A. f

B. f

C. f

D. f

−1

−1

−1

−1

(x) =

, for x ≠

(x) =

, for x ≥

(x) =

, for x ≤

(x) =

, for all x

b. Verify that the equation is correct.

f f

−1

(x)

= f

and

=

f

−1

(f(x)) = f

=

−1

Substitute.

Simplify.

The equation is (1)

(1)

not verified.

verified.

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Homework 9 (for Test 3)-ariel gonzalez

3/16/24, 15:01

19. Watch the video and then solve the problem given below.

Click here to watch the video.2

The function f(x) =

x + 12

is one-to-one. For the function,

x−3

a. Find an equation for f

−1

(x), the inverse function.

b. Verify that your equation is correct by showing that f f

−1

(x) = x and f

−1

(f(x)) = x.

a. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

(Simplify your answer. Use integers or fractions for any numbers in the expression.)

A. f

B. f

C. f

D. f

−1

−1

−1

−1

(x) =

, for all x

(x) =

, for x ≤

(x) =

, for x ≥

(x) =

, for x ≠

Verify that the equation is correct.

f f

−1

(x ) = f

and

=

and

f

−1

(f(x)) = f

−1

=

Substitute.

Simplify.

The equation is (1)

2: http://https://mediaplayer.pearsoncmg.com/assets/bzca7e_02_07_02

(1)

verified.

not verified.

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Homework 9 (for Test 3)-ariel gonzalez

3/16/24, 15:01

20. Select Function B and check the Display Options f(x), f− 1 (x) and both Show Eq. boxes. Complete parts 1 through 4 below.

Use the interactive figure to find your answer. Use the left and right arrow keys to move along a slider as needed.

Click here to launch the interactive figure.3

Part 1: What is the domain of f(x)?

The domain of f(x) is

. (Type your answer in interval notation.)

Part 2: What is the range of f(x)?

The range of f(x) is

. (Type your answer in interval notation.)

Part 3: What is the domain of f

The domain of f

−1

−1

(x)?

(x) is

Part 4: What is the range of f

The range of f

−1

(x) is

. (Type your answer in interval notation.)

−1

(x)?

. (Type your answer in interval notation.)

3: http://media.pearsoncmg.com/cmg/pmmg_mml_shared/precalc_ifigs_HTML5/IFig1_28Precalc_graphs_of_inverse_functions/inde

*21. Solve.

(x − 5)(x + 4)

0

x−5

y

4

2

The solution set is

.

(Type your answer in interval notation.)

x

(3,0)

-2

2

4

6

8

-2

y=

-4

10

12

x−3

x−5

-6

31. Given the function f(x) = 6x − 4.

(a) Find f

−1

.

(b) Graph f and f

−1

in the same rectangular coordinate system.

(c) Use interval notation to give the domain and the range of f and f

−1

.

−1

(a) The inverse function is f (x) =

.

(Use integers or fractions for any numbers in the expression.)

(b) Choose the correct graph which shows f(x) and f

line and the graph of f

−1

−1

(x) graphed in the same coordinate system. The graph of f(x) is shown as a solid

(x) is shown as a dotted line.

A.

B.

10

y

C.

y

10

x

-10

10

D.

10

x

-10

-10

y

10

x

10

-10

-10

y

x

10

-10

-10

10

-10

(c) State the domain and range of f(x) using interval notation.

The domain of f(x) is

State the domain and range of f

The domain of f

−1

(x) is

, and the range of f(x) is

−1

.

(x) using interval notation.

, and the range of f

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−1

(x) is

.

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Homework 9 (for Test 3)-ariel gonzalez

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32. Given the function f(x) = x2 − 19, x ≥ 0, complete parts a through c.

(a) Find an equation for f

(b) Graph f and f

−1

−1

(x).

in the same rectangular coordinate system.

(c) Use interval notation to give the domain and the range of f and f

(a) Find f

−1

−1

.

(x).

−1

f (x) =

(Type an exact answer, using radicals as needed.)

(b) Graph f and f

−1

in the same coordinate system. Choose the correct graph below.

A.

B.

30

y

C.

30

x

-30

30

-30

-30

The domain of f(x) is

−1

(x) is

y

30

x

30

−1

30

-30

30

-30

using interval notation.

, and the range of f(x) is

, and the range of f

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y

x

-30

(c) State the domain and range of f and f

The domain of f

30

x

-30

-30

D.

y

−1

.

(x) is

.

Page 14 of 19

Homework 9 (for Test 3)-ariel gonzalez

3/16/24, 15:01

33. Given the function f(x) = x3 − 12, complete parts a through c.

(a) Find an equation for f

(b) Graph f and f

−1

−1

(x).

in the same rectangular coordinate system.

(c) Use interval notation to give the domain and the range of f and f

(a) Find f

−1

−1

.

(x).

−1

f (x) =

(Type an exact answer, using radicals as needed.)

(b) Graph f and f

−1

in the same coordinate system. Choose the correct graph below.

A.

B.

18

y

C.

18

x

-18

18

-18

-18

The domain of f(x) is

−1

(x) is

y

18

x

18

−1

18

-18

18

-18

using interval notation.

, and the range of f(x) is

, and the range of f

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y

x

-18

(c) State the domain and range of f and f

The domain of f

18

x

-18

-18

D.

y

−1

.

(x) is

.

Page 15 of 19

Homework 9 (for Test 3)-ariel gonzalez

34.

Given the function f(x) =

3

(a) Find an equation for f

(b) Graph f and f

−1

3/16/24, 15:01

x + 4, complete parts a through c.

−1

(x).

in the same rectangular coordinate system.

(c) Use interval notation to give the domain and the range of f and f

−1

.

n

(Hint: To solve for a variable involving an nth root, raise both sides of the equation to the nth power,

a) Find f

−1

n

y

= y.)

(x). Select the correct choice below and fill in the answer box(es) to complete your choice.

(Simplify your answer. Use integers or fractions for any numbers in the expression.)

A. f

B. f

C. f

D. f

−1

−1

−1

−1

(x) =

, x≠

(x) =

, x≥

(x) =

, x≤

(x) =

, for all x

b) Graph f and f

−1

in the same rectangular coordinate system. Choose the correct graph below.

A.

B.

8

y

C.

y

8

8

x

-8

-8

-8

8

8

-8

The domain of f(x) is

8

−1

-8

8

-8

using interval notation.

, and the range of f(x) is

(x) is

x

-8

(c) State the domain and range of f and f

y

x

-8

−1

y

x

8

The domain of f

D.

, and the range of f

−1

.

(x) is

.

35. Select Function D and check the Display Options f(x), f− 1 (x) and both Show Eq. boxes. If the graph of f(x) has a vertical asymptote

x = 3, then what is the equation of the horizontal asymptote of f

−1

(x)?

Use the interactive figure to find your answer. Use the left and right arrow keys to move along a slider as needed.

Click here to launch the interactive figure.5

The horizontal asymptote of f

(Type an equation.)

−1

(x) is described by the equation

.

5: http://media.pearsoncmg.com/cmg/pmmg_mml_shared/precalc_ifigs_HTML5/IFig1_28Precalc_graphs_of_inverse_functions/inde

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Homework 9 (for Test 3)-ariel gonzalez

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36. The functions f and g are defined by the following tables. Use the

tables to evaluate the given composite function.

f(g(10))

x

f(x)

x

g(x)

3

−2

−3

−5

4

0

3

1

5

1

5

2

−6

4

10

3

f(g(10)) =

*37. Match the following function with its graph.

y=

1

2

(x − 6) + 4

4

Choose the correct graph below.

A.

B.

10

y

C.

10

x

-10

10

-10

y

D.

10

y

10

x

-10

10

y

x

-10

-10

38. The functions f and g are defined by the following tables. Use the

tables to evaluate the given composite function.

(g ◦ f)( − 2)

x

10

-10

-10

10

-10

x

f(x)

x

g(x)

−2

3

−4

−6

0

4

3

−4

1

5

4

2

5

−1

9

−1

x

f(x)

x

g(x)

−2

3

−2

0

0

4

3

−4

(g ◦ f)( − 2) =

39. The functions f and g are defined by the following tables. Use the

tables to evaluate the given composite function.

f

f

−1

−1

(g(10))

1

5

4

2

5

−5

10

3

(g(10)) =

40. Let the functions f and g be given by the equations on the right.

Evaluate the indicated function without finding an equation for the

function.

f(x) = 8x − 1

g(x) = 2x − 5

(f ◦ g)(3)

(f ◦ g)(3) =

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Homework 9 (for Test 3)-ariel gonzalez

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*41. Solve the inequality. State the solution set in interval notation and graph it.

2

4x + 21 < x
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is
.
(Type your answer in interval notation.)
B. The solution is all real numbers.
C. The solution is the empty set.
Graph the solution set. Choose the correct answer below.
A.
B.
-10
-5
0
5
10
C.
-10
-5
0
5
10
-10
-5
0
5
10
-10
-5
0
5
10
D.
-10
-5
0
5
10
E.
F.
-10
-5
0
5
10
42. Let the function f be given by the equation f(x) = 2x − 8. Evaluate f− 1 (2) without finding an equation for the function f− 1 (x).
f
−1
(2) =
43. Determine whether the following statement makes sense or does not make sense, and explain your reasoning.
A student found the inverse of f(x) = 3x − 2 mentally by reversing "multiplying by 3 and subtracting 2" to get "adding 2 and dividing by 3,"
x+2
−1
so f
(x) =
.
3
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The statement makes sense because f− 1 undoes the changes produced by f. In this case, the first
operation on x is multiplication and the second is subtraction. So, first undo subtraction then multiplication.
B. The statement does not make sense because f− 1 is the reciprocal f. Hence, f− 1 (x) =
(Use integers or fractions for any numbers in the expression.)
.
C. The statement does not make sense because f− 1 reverses the changes produced by f. In this case, the
first operation on x is multiplication and the second is subtraction. So, first perform subtraction and then
−1
multiplication. Hence, f (x) =
.
(Use integers or fractions for any numbers in the expression.)
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Homework 9 (for Test 3)-ariel gonzalez
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44. Determine whether the following statement is true or false. If the statement is false, make the necessary change(s) to produce a true
statement.
The inverse of {(2,3),(3,5)} is {(3,5),(2,3)}.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The statement is true.
B. The statement is false. The correct statement is the inverse of {(2,3),(3,5)} is
(Type an ordered pair. Use a comma to separate answers as needed.)
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.
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