Deck 11: Additional Properties of Functions

Question 1

Evaluate the function.
If $g ( x ) = 4 x - 2$ , find $g ( a - 1 )$

A) $4 a - 8$
B) $4 a - 6$
C) $4 a - 4$
D) $4 a - 3$

Answer 1

B

Question 2

Solve
A wind turbine generator produces P kilowatts of power for wind speed w (measured in miles per hour) according to the equation $\mathrm { P } = 2.6 \mathrm { w } ^ { 2 }$ . Thus, the number of kilowatt hours $\mathrm { P }$ as a function of $\mathrm { w }$ is $\mathrm { P } ( \mathrm { w } ) = 2.6 \mathrm { w } ^ { 2 }$ . At a particular time the wind speed is measured to be 20 miles per hour. However, the measuring device is known to have an inherent error of measure, e. Find an expression for the actual amount of energy produced by finding $\mathrm { P } ( 20 + \mathrm { e } )$ .

A) $\mathrm { P } ( 20 + \mathrm { e } ) = 2.6 \mathrm { e } ^ { 2 } + 20$
B) $\mathrm { P } ( 20 + \mathrm { e } ) = 2.6 \mathrm { e } ^ { 2 } + 1040$
C) $\mathrm { P } ( 20 + \mathrm { e } ) = 2 \cdot 6 \mathrm { e } ^ { 2 } + 104 \mathrm { e } + 1040$
D) $\mathrm { P } ( 20 + \mathrm { e } ) = \mathrm { e } + 1040$

Answer 2

C

Question 3

Evaluate the function.
If $f ( x ) = - 9 x + 4$ , find $f ( - 3 )$

A) 31
B) $- 5$
C) 23
D) 15

Answer 3

A

Question 4

Determine whether or not the graph represents a function.

A)not a function
B)function

Question 5

Fill in the blank.
To obtain the graph of f(x - k), shift the of f(x)k units.

A)to the left
B)up
C)to the right
D)down

Question 6

Evaluate the function.
If $r ( x ) = \frac { 3 } { x - 4 }$ , find $r ( a + 5 )$

A) $- \frac { 3 } { 4 } a - \frac { 15 } { 4 }$
B) $\frac { 3 } { a - 4 } + 3$
C) $\frac { 8 } { a - 4 }$
D) $\frac { 3 } { a + 1 }$

Question 7

Evaluate the function.
If $f ( x ) = x ^ { 2 } - 2$ , find $f ( a - 3 )$ .

A) $a ^ { 2 } - 6 a + 9$
B) $a ^ { 2 } - 5$
C) $a ^ { 2 } - 6 a + 7$
D) $a ^ { 2 } + 9$

Question 8

Evaluate the function.
If $f ( x ) = 6 x - 11$ , find $f ( 0 )$

A) 0
B) $- 11$
C) 6
D) $- 5$

Question 9

Fill in the blank.
To obtain the graph of f(x)- k, shift the of f(x)k units.

A)up
B)to the left
C)to the right
D)down

Question 10

Determine whether or not the graph represents a function.

A)not a function
B)function

Question 11

Determine whether or not the graph represents a function.

A)not a function
B)function

Question 12

Evaluate the function.
If $h ( x ) = \sqrt { x - 7 }$ , find $h \left( a ^ { 2 } + 17 \right)$

A) $a + 10$
B) $\sqrt { a ^ { 2 } + 10 }$
C) $a + \sqrt { 10 }$
D) $\mathrm { a } + 100$

Question 13

Evaluate the function.
If $f ( x ) = 5 x ^ { 2 } + 3 x - 4$ , find $f ( k - 1 )$

A) $- 7 k ^ { 2 } + 5 k - 2$
B) $5 k ^ { 2 } - 17 k + 4$
C) $5 \mathrm { k } ^ { 2 } - 7 \mathrm { k } + 4$
D) $5 \mathrm { k } ^ { 2 } - 7 \mathrm { k } - 2$

Question 14

Evaluate the function.
If $h ( x ) = 2 x - 9$ , find $h \left( a ^ { 2 } \right) - h \left( \frac { 1 } { 3 } \right)$

A) $2 a ^ { 2 } - \frac { 29 } { 3 }$
B) $2 a ^ { 2 } - \frac { 2 } { 3 }$
C) $2 a ^ { 2 } - \frac { 2 } { 3 } x - 9$
D) $2 a ^ { 2 } - \frac { 56 } { 3 }$

Question 15

Evaluate the function.
If $r ( x ) = \frac { 6 } { x - 4 }$ , find $r ( - 3 )$

A) $- \frac { 3 } { 4 }$
B) 6
C) $\frac { 9 } { 2 }$
D) $- \frac { 6 } { 7 }$

Question 16

Solve
A rope 24 feet long is cut into two unequal pieces. Each piece is used to form a square. Let x represent the length of the shorter piece of rope. Write a function $\mathrm { A } ( \mathrm { x } )$ that expresses the total area enclosed by the two squares.

A) $A ( x ) = 2 x ^ { 2 } - 48 x + 576$
B) $A ( x ) = \frac { 2 x ^ { 2 } - 48 x + 576 } { 16 }$
C) $A ( x ) = 576 - 48 x$
D) $A ( x ) = \frac { 576 - x ^ { 2 } } { 16 }$

Question 17

Evaluate the function.
If $f ( x ) = \sqrt { x + 6 }$ , find $f ( 4 a + 2 )$

A) $\sqrt { 4 a + 6 } + 2 \sqrt { 2 }$
B) $\sqrt { 4 a + 10 }$
C) $2 \sqrt { a + 8 }$
D) $2 \sqrt { a + 2 }$

Question 18

Evaluate the function.
If $g ( x ) = 5 x - 3$ , find $g ( a ) + g ( - 1 )$

A) $5 \mathrm { a } - 11$
B) $5 a - 5$
C) $5 a - 4$
D) $5 a - 8$

Question 19

Evaluate the function.
If $g ( x ) = \frac { 1 } { 3 } x - 2$ , find $g ( a + 1 )$

A) $\frac { a + 5 } { 3 }$
B) $\frac { a - 5 } { 3 }$
C) $\frac { a - 11 } { 3 }$
D) $\frac { a - 3 } { 3 }$

Question 20

Evaluate the function.
If $p ( x ) = 4 x ^ { 2 } + 3 x - 1$ , find $p \left( - \frac { a ^ { 2 } } { 4 } \right)$

A) $- a ^ { 4 } - 3 a ^ { 2 } - \frac { 1 } { 4 }$
B) $a ^ { 4 } - \frac { 3 a ^ { 2 } } { 4 } - 1$
C) $\frac { a ^ { 4 } } { 4 } - \frac { 3 a ^ { 2 } } { 4 } - 1$
D) $\frac { - a ^ { 4 } - 3 a ^ { 2 } - 1 } { 4 }$

Question 21

Perform the requested operation on the given function.
$$f ( x ) = 2 x - 8 , g ( x ) = 6 x - 5$$
Find $( g - f ) ( x )$ .

A) $x - 13$
B) $- 4 x - 13$
C) $x + 3$
D) $- 4 x - 3$

Question 22

Determine whether or not the graph represents a function.

A)function
B)not a function

Question 23

Determine whether or not the graph represents a function.

A)not a function
B)function

Question 24

Perform the requested operation on the given function.
$$f ( x ) = 2 x - 8 , g ( x ) = 8 x - 3$$
Find $( f - g ) ( x )$ .

A) $10 x - 11$
B) $- 6 x - 11$
C) $x + 5$
D) $- 6 x - 5$

Question 25

Graph the two functions on one coordinate plane.
$\begin{array}{l}f(x)=|x| \\5(x)=|x|-5\end{array}$

A)
B)
C)
D)

Question 26

Determine whether or not the graph represents a function.

A)function
B)not a function

Question 27

Graph the two functions on one coordinate plane.
$\begin{array}{l}f(x)=x^{2} \\g(x)=(x+3)^{2}-5\end{array}$

A)
B)
C)
D)

Question 28

Perform the requested operation on the given function.
$$f ( x ) = - 2 x ^ { 2 } + 6 , g ( x ) = x - 2$$
Find $( f - g ) ( - 4 )$ .

A)
B) $- 20$
C) $- 28$
D) $- 24$

Question 29

Graph the two functions on one coordinate plane.
$$f ( x ) = \frac { 3 } { x }$$
$$g ( x ) = \frac { 3 } { x } + 4$$

A)
B)
C)
D)

Question 30

Perform the requested operation on the given function.
$$f ( x ) = x + 3 , g ( x ) = x - 4$$
Find $( f + g ) ( - 5 )$ .

A) $- 3$
B) $- 11$
C) $- 17$
D) $- 9$

Question 31

Determine whether or not the graph represents a function.

A)not a function
B)function

Question 32

Determine whether or not the graph represents a function.

A)function
B)not a function

Question 33

Graph the two functions on one coordinate plane.
$\begin{array}{l}f(x)=x^{2} \\h(x)=x^{2}-2\end{array}$

A)
B)
C)
D)

Question 34

Graph the two functions on one coordinate plane.
$\begin{array}{l}(x)=x^{3} \\5(x)=x^{3}+3\end{array}$

A)
B)
C)
D)

Question 35

Determine whether or not the graph represents a function.

A)not a function
B)function

Question 36

Graph the two functions on one coordinate plane.
$\begin{array}{l}(x)=|x| \\y(x)=|x-4|+1\end{array}$

A)
B)
C)
D)

Question 37

Graph the two functions on one coordinate plane.
$\begin{array}{l}(x)=x^{3} \\(x)=(x+5)^{3}+7\end{array}$

A)
B)
C)
D)

Question 38

Perform the requested operation on the given function.
$$f ( x ) = 2 - 9 x , g ( x ) = - 6 x + 9$$
Find $( f + g ) ( x )$ .

A) $- 6 x + 2$
B) $- 15 x + 11$
C) $- 3 x + 11$
D) $- 4 x$

Question 39

Determine whether or not the graph represents a function.

A)function
B)not a function

Question 40

Graph the two functions on one coordinate plane.
$\begin{array}{l}(x)=x^{2} \\(x)=(x-2)^{2}\end{array}$

A)
B)
C)
D)

Question 41

Find the composition.
If $f ( x ) = 6 x + 2$ and $g ( x ) = x + 4$ , find $( g \circ f ) ( x )$

A) $6 x + 6$
B) $6 x ^ { 2 } + 26 x + 8$
C) $6 x + 26$
D) $6 x ^ { 2 } + 26$

Question 42

Find the composition.
If $f ( x ) = x ^ { 2 } - 1$ and $h ( x ) = \sqrt { x + 7 }$ , find $( h \circ f ) ( - 3 )$

A)
B) $\sqrt { 15 }$
C) $- 3 \sqrt { 6 }$
D) $2 + \sqrt { 7 }$

Question 43

Find the composition.
If $f ( x ) = \frac { 3 } { x + 5 }$ and $g ( x ) = \frac { 1 } { 7 x }$ , find $f [ g ( x ) ]$

A) $\frac { 1 x + 5 } { 21 x }$
B) $\frac { 21 x } { 1 + 35 x }$
C) $\frac { 3 x } { 1 + 35 x }$
D) $\frac { 21 x } { 1 - 35 x }$

Question 44

Perform the requested operation on the given function.
$$f ( x ) = x + 3 , g ( x ) = 5 x ^ { 2 } + 13 x + 6$$
Find $( f g ) ( - 4 )$ .

A)
B) $- 34$
C) $- 238$
D) $- 602$

Question 45

Find the composition.
If $f ( x ) = 4 x ^ { 2 } + 2 x + 4$ and $g ( x ) = 2 x - 7$ , find $g [ f ( x ) ]$

A) $4 x ^ { 2 } + 4 x + 1$
B) $8 x ^ { 2 } + 4 x + 15$
C) $8 x ^ { 2 } + 4 x + 1$
D) $4 x ^ { 2 } + 2 x - 3$

Question 46

Find the composition.
If $f ( x ) = \frac { x - 6 } { 4 }$ and $g ( x ) = 4 x + 6$ , find $g [ f ( x ) ]$

A) $x$
B) $x - \frac { 3 } { 2 }$
C) $x + 12$
D) $4 x + 18$

Question 47

Perform the requested operation on the given function.
$$f ( x ) = x ^ { 2 } - \frac { 4 } { 5 } x - 4 , g ( x ) = x ^ { 3 } + \frac { 3 } { 5 } x ^ { 2 } + x$$
Find $( f + g ) ( x )$

A) $2 x ^ { 3 } - \frac { 1 } { 10 } x ^ { 2 } - 3 x$
B) $2 x ^ { 3 } - \frac { 1 } { 5 } x ^ { 2 } - 3 x$
C) $x ^ { 3 } + \frac { 4 } { 5 } x ^ { 2 } + \frac { 1 } { 10 } x - 4$
D) $x ^ { 3 } + \frac { 8 } { 5 } x ^ { 2 } + \frac { 1 } { 5 } x - 4$

Question 48

Perform the requested operation on the given function.
$$f ( x ) = 5 x + 2 , g ( x ) = 3 x ^ { 2 } + 4 x + 4$$
Find $\left( \frac { f } { g } \right) ( 5 )$ .

A)ـــــ
B) $-$
C)ــــــ
D) $- \frac { 3 } { 11 }$

Question 49

Find the composition.
If $f ( x ) = \sqrt { x + 9 }$ and $g ( x ) = 8 x - 13$ , find $f [ g ( x ) ]$

A) $2 \sqrt { 2 x - 1 }$
B) $8 \sqrt { x - 4 }$
C) $2 \sqrt { 2 x + 1 }$
D) $8 \sqrt { x + 9 } - 13$

Question 50

Perform the requested operation on the given function.
$$f ( x ) = \sqrt { 2 x } , g ( x ) = - 5 x + 8$$
Find $( f g ) ( x )$ .

A) $- 5 x \sqrt { 2 x } + 8 \sqrt { 2 x }$
B) $- 10 x ^ { 2 } + 16$
C) $- 10 x + 8 \sqrt { 2 x }$
D) $\sqrt { 2 \mathrm { x } } + 40 \mathrm { x }$

Question 51

Perform the requested operation on the given function.
$$f ( x ) = x ^ { 2 } + 10 x + 24 , g ( x ) = x + 6$$
Find $\left( \frac { \mathrm { f } } { \mathrm { g } } \right) ( \mathrm { x } )$ .

A) $\frac { 1 } { x + 4 } , x \neq 4$
B) $\frac { 1 } { x + 4 } , x \neq 4$ and $x \neq 6$
C) $x + 4 , x \neq$
D) $x + 4 , x \neq$

Question 52

Perform the requested operation on the given function.
$$f ( x ) = 6 x - 7 , g ( x ) = 3 x - 5$$
Find $( \mathrm { fg } ) ( \mathrm { x } )$ .

A) $18 x ^ { 2 } + 35$
B) $18 x ^ { 2 } - 26 x + 35$
C) $9 x ^ { 2 } - 51 x - 12$
D) $18 x ^ { 2 } - 51 x + 35$

Question 53

Find the composition.
If $f ( x ) = 6 x ^ { 2 } - 5 x$ and $g ( x ) = - 3 x$ , find $( g \circ f ) ( x )$

A) $- 18 x ^ { 3 } + 15 x ^ { 2 }$
B) $- 18 x ^ { 2 } + 15 x$
C) $6 x ^ { 2 } - 10$
D) $54 x ^ { 2 } - 5 x$

Question 54

Find the composition.
If $f ( x ) = x ^ { 2 } - 6$ and $h ( x ) = \sqrt { x + 13 }$ , find $( h \circ f ) ( - 4 )$

A) $- 2 + \sqrt { 13 }$
B) 3
C) $- 4 \sqrt { 7 }$
D) $\sqrt { 23 }$

Question 55

Perform the requested operation on the given function.
$$f ( x ) = \sqrt { 2 x + 3 } , g ( x ) = \sqrt { 4 x - 25 }$$
Find $( f g ) ( x )$ .

A) $( 2 x - 5 ) ( \sqrt { 2 x + 3 } )$
B) $( 2 x + 3 ) ( 4 x - 25 )$
C) $( 2 x + 3 ) ( 2 x - 5 )$
D) $( \sqrt { 2 x + 3 } ) ( \sqrt { 4 x - 25 } )$

Question 56

Find the composition.
If $f ( x ) = 2 x + 5$ and $g ( x ) = x + 5$ , find $( f \circ g ) ( x )$

A) $2 x ^ { 2 } + 15$
B) $2 x + 10$
C) $2 x ^ { 2 } + 15 x + 25$
D) $2 x + 15$

Question 57

Perform the requested operation on the given function.
$$f ( x ) = 2 x ^ { 2 } - 7 x - 6 , g ( x ) = - 7 x + 2$$
Find $( f + g ) ( - 4 )$ .

A) $- 56$
B) 84
C) $- 66$
D)

Question 58

Find the composition.
If $f ( x ) = - 3 x - 4$ and $g ( x ) = \frac { 1 } { x }$ , find $g \left[ f \left( \frac { 1 } { 4 } \right] \right.$ .

A) $- \frac { 19 } { 4 }$
B) $- 16$
C) $- \frac { 4 } { 19 }$
D) $- \frac { 1 } { 19 }$

Question 59

Find the composition.
If $f ( x ) = 7 x ^ { 2 } + 4 x$ and $g ( x ) = 2 x$ , find $( f \circ g ) ( x )$

A) $7 x ^ { 2 } + 6$
B) $28 x ^ { 2 } + 8 x$
C) $14 x ^ { 3 } + 8 x ^ { 2 }$
D) $14 x ^ { 2 } + 8 x$

Question 60

Perform the requested operation on the given function.
$$f ( x ) = 2 x ^ { 2 } - 7 x , g ( x ) = x ^ { 2 } - 4 x - 21$$
Find $\left( \frac { f } { g } \right) ( x )$ .

A) $\frac { 2 x - 7 } { - 4 }$
B) $\frac { 2 x } { x + 1 }$
C) $\frac { 2 - x } { 21 }$
D) $\frac { 2 x ^ { 2 } - 7 x } { x ^ { 2 } - 4 x - 21 }$

Question 61

Perform the requested operation on the given function.
$$f ( x ) = 4 \sqrt { x - 6 } , g ( x ) = 2 \sqrt { x - 6 }$$
Find $( f + g ) ( x )$

A) $6 x + 6$
B) 6
C) $6 \sqrt { x - 6 }$
D) $6 \sqrt { 2 x - 12 }$

Question 62

Indicate whether the function is one-to-one.
An oil well off the Gulf Coast is leaking, with the leak spreading oil over the surface of the gulf as a circle. At any time $t$ , in minutes, after the beginning of the leak, the radius of the oil slick on the surface is $r ( t ) = 4 t$ ft. Find the area A of the oil slick as a function of time.
A) $A ( r ( t ) ) = 16 \pi t ^ { 2 }$
B) $\mathrm { A } ( \mathrm { r } ( \mathrm { t } ) ) = 16 \mathrm { t } ^ { 2 }$
C) $\mathrm { A } ( \mathrm { r } ( \mathrm { t } ) ) = 4 \pi \mathrm { t } ^ { 2 }$
D) $A ( r ( t ) ) = 16 \pi t$

A)Yes
B)No

Question 63

Solve.
$$A = \{ ( - 5,16 ) , ( 16,16 ) , ( 2 , - 16 ) \}$$

Question 64

Indicate whether the graph represents a one-to-one function

A)Yes
B)No

Question 65

Indicate whether the graph represents a one-to-one function

A)Yes
B)No

Question 66

Indicate whether the function is one-to-one.
$$C = \{ ( 6 , - 9 ) , ( 2 , - 8 ) , ( 0 , - 7 ) , ( - 2 , - 6 ) \}$$

A)Yes
B)No

Question 67

Indicate whether the function is one-to-one.
The surface area of a balloon is given by $S ( r ) = 4 \pi r ^ { 2 }$ , where $r$ is the radius of the balloon. If the radius is increasing with time $t$ , as the balloon is being blown up, according to the formula $r ( t ) = \frac { 4 } { 5 } t ^ { 3 } , t \geq 0$ , find the surface area $S$ as a function of the time $t$ .
A) $S ( r ( t ) ) = \frac { 64 } { 25 } \pi t ^ { 9 }$
B) $S ( r ( t ) ) = \frac { 16 } { 25 } \pi t ^ { 6 }$
C) $\mathrm { S } ( \mathrm { r } ( \mathrm { t } ) ) = \frac { 64 } { 25 } \pi \mathrm { t } ^ { 6 }$
D) $\mathrm { S } ( \mathrm { r } ( \mathrm { t } ) ) = \frac { 64 } { 25 } \pi \mathrm { t } ^ { 3 }$

A)Yes
B)No

Question 68

Indicate whether the graph represents a one-to-one function

A)Yes
B)No

Question 69

Solve.
$$B = \{ ( 4,2 ) , ( 5,2 ) , ( 6,8 ) , ( 7,4 ) \}$$

Question 70

Indicate whether the function is one-to-one.
Suppose the dollar cost to produce $n$ items in a factory is $c ( n ) = 4 n + 3$ . Furthermore, the number of items $n$ produced in $x$ hours is $n ( x ) = 6 x$ . Find $c [ n ( x ) ]$ , which is the composite function that defines the dollar cost in terms of the number of hours of production $x$ .
A) $c [ n ( x ) ] = 10 x + 18$
B) $c [ n ( x ) ] = 24 x + 3$
C) $c [ n ( x ) ] = 24 x + 18$
D) $c [ n ( x ) ] = 10 x + 3$

A)Yes
B)No

Question 71

Indicate whether the graph represents a one-to-one function

A)Yes
B)No

Question 72

Indicate whether the graph represents a one-to-one function

A)Yes
B)No

Question 73

Solve.
$$C = \{ ( 7 - 5 ) , ( 5 - 7 ) , ( 2,3 ) , ( - 2 , - 3 ) \}$$

Question 74

Solve.
$$A = \{ ( 2,17 ) , ( 17,13 ) , ( 3 , - 18 ) \}$$

Question 75

Indicate whether the function is one-to-one.
The population $\mathrm { P }$ of a predator mammal depends upon the number $\mathrm { x }$ of a smaller animal that is its primary food source. The population s of the smaller animal depends upon the amount a of a certain plant that is its primary food source. If $\mathrm { P } ( \mathrm { x } ) = 3 \mathrm { x } ^ { 2 } + 8$ and $s ( a ) = 4 a + 4$ , what is the relationship between the predator mammal and the plant food source?
A) $\mathrm { P } ( \mathrm { s } ( \mathrm { a } ) ) = 48 \mathrm { a } ^ { 2 } + 96 \mathrm { a } + 56$
B) $\mathrm { P } ( \mathrm { s } ( \mathrm { a } ) ) = 48 \mathrm { a } ^ { 2 } + 48 \mathrm { a } + 56$
C) $\mathrm { P } ( \mathrm { s } ( \mathrm { a } ) ) = 16 \mathrm { a } ^ { 2 } + 32 \mathrm { a } + 24$
D) $\mathrm { P } ( \mathrm { s } ( \mathrm { a } ) ) = 12 \mathrm { a } + 12$

A)Yes
B)No

Question 76

Find the composition.
If $f ( x ) = x ^ { 2 } + 2 x$ and $g ( x ) = x + 4$ , find $( f \circ g ) ( 4 )$

A) 0
B) $2 \sqrt { 3 }$
C) 28
D) 80

Question 77

Indicate whether the graph represents a one-to-one function

A)Yes
B)No

Question 78

Find the composition.
If $f ( x ) = x ^ { 2 } + 4 x$ and $g ( x ) = x + 2$ , find $( g \circ f ) ( 4 )$

A) 192
B) 60
C) 34
D) 36