Deck 2: Functions

Question 1

Find the domain of the function .

Answer 1

There is no value of that makes the denominator , thus the domain is all real numbers . 

Question 2

Determine if the equation defines as a function of . Explain your answer.

Answer 2

or . No. Since this equation gives two values of for a given value of , the given equation does not define a function of .

Question 3

Complete the table for the function .

Answer 3

x

Question 4

Express the rule in function notation. Multiply by $4$ , then subtract $2$ .

A) $f ( x ) = 2 x + 4$
B) $f ( x ) = - 18 x$
C) $f ( x ) = 4 x - 2$
D) $f ( x ) = 4 ( x + 2 )$
E) $f ( x ) = 4 ( x - 2 )$

Question 5

Let .
(a) Sketch the graph of .
(b) Find the domain of .
(c) State the intervals on which is increasing and on which is decreasing.

Question 6

If , find , , , , and .

Question 7

Evaluate , , , , and for the piecewise-defined function.

Question 8

For the function , find and .

Question 9

Express the rule in function notation.
Square, subtract $5$ , then take the square root.

A)
$f ( x ) = \sqrt { x ^ { 2 } - 5 }$
B)
$f ( x ) = x - \sqrt { 5 }$
C)
$f ( x ) = x ^ { 2 } - 5$
D)
$f ( x ) = \sqrt { 5 x ^ { 2 } }$
E)
$f ( x ) = \sqrt { - 5 x ^ { 2 } }$

Question 10

Find the domain of the function .

Question 11

State whether the curve is the graph of a function. If it is, state the domain and range of the function.

A) Function, domain $$[ - 2,2 )$$ , range
$( - 4,4 )$
B) Function, domain $[ - 2,2 ]$ , range
$[ - 4,4 ]$
C) Function, domain $( - 2,2 )$ , range
$( - 4,4 )$
D) Function, domain $( - 4,4 )$ , range
$( - 2,2 )$
E) Not a function

Question 12

Determine if the equation defines as a function of . Explain your answer.

Question 13

If , find , , .

Question 14

If , find , , .

Question 15

If , find , , .

Question 16

Let .
(a) Sketch the graph of .
(b) Find the domain and the range of .

Question 17

Determine if the equation defines as a function of . Explain your answer.

Question 18

Determine whether the given curve is the graph of a function of $x$ . If it is, state the domain and range of the function.

A) Function, domain: $( - \infty , - 2 ) \cup ( - 2 , \infty )$ , range:
$[ 2 , \infty )$
B) Function, domain: $( - \infty , - 2 ) \cup ( 2 , \infty )$ , range:
$[ - 2 , \infty )$
C) Function, domain: $( - \infty , 0 )$ , range:
$[ 2 , \infty )$
D) Function, domain: $( - \infty , - 2 ) \cup ( 0 , \infty )$ , range:
$[ - 2 , \infty )$
E) Not a function

Question 19

The cost in dollars to manufacture integrated circuits is given by the function , where is measured in hundreds.
(a) Find and .
(b) What do the answers in part (a) represent?

Question 20

Determine whether the given curve is the graph of a function of $x$ . If it is, state the domain and range of the function.

A) Function, domain: $[ 0 , \infty )$ , range:
$( - \infty , \infty )$
B) Function, domain: $( - \infty , \infty )$ range:
$[ 0 , \infty )$
C) Function, domain: $( 0 , \infty )$ , range:
$( - \infty , \infty )$
D) Function, domain: $( - \infty , \infty )$ , range:
$( - \infty , \infty )$
E) Not a function

Question 21

Use a graphing device to the graph of the function . State approximately the intervals on which the function is increasing and on which the function is decreasing.

Question 22

Sketch the graph of the piecewise-defined function.

Question 23

Let . Graph the family of functions with , , , , and in the viewing rectangle by . How does the value of affect the graph?

Question 24

Use a graphing device to draw the graph of the function . State approximately the intervals on which the function is increasing and on which the function is decreasing.

Question 25

Let . Graph the family of functions with , , , , and in the viewing rectangle by . How does the value of affect the graph?

Question 26

Graphs of the functions f and g are given.
(a) Which is larger,
(b) Which is larger,
(c) For which values of x is

Question 27

The graph of a function h is given.
(a) Find
(b) Find the domain and range of h.
(c) Find the values of x for which
(d) Find the values of x for which

Question 28

The graph of a function is given. Find the intervals on which the function is increasing and decreasing.

A) Increasing on
$[ - 1,1 ]$ ;
$[ 3 , \infty )$ ; decreasing on
$( - \infty , - 1 ]$ ,
$[ 1,3 ]$
B) Increasing on
$$[ - 1,1 ]$$ ; decreasing on
$( - \infty , - 1 ]$
C) Increasing on
$[ - 4,1 ]$ ; decreasing on
$[ 0 , - 4 ]$
D) Increasing on
$[ - 4,1 ]$ ; decreasing on
$[ - 4,0 ]$
E) Increasing on
$[ - 1,1 ]$ ,
$[ 3 , \infty )$ ; decreasing on
$[ 6 , - 4 ]$ ,
$[ - 4,0 ]$

Question 29

A function is given. Use a graphing calculator to draw the graph of f. Find the domain and range of f from the graph.

Question 30

Use a graphing device to draw the graph of the function . State approximately the intervals on which the function is increasing and on which the function is decreasing.

Question 31

A function is given. Use a graphing calculator to draw the graph of f. Find the domain and range of f from the graph.

Question 32

A function is given. Use a graphing calculator to draw the graph of f. Find the domain and range of f from the graph.

Question 33

A function is given. Use a graphing calculator to draw the graph of f. Find the domain and range of f from the graph.

Question 34

A function is given.
(a) Find all the local maximum and minimum values of the function and the value of x at which each occurs.
(b) Find the intervals on which the function is increasing and on which the function is decreasing. State all answers correct to two decimal places.

Question 35

Use a graphing device to draw the graph of the function . State approximately the intervals on which the function is increasing and on which the function is decreasing.

Question 36

Let . Graph the family of functions with , , and in the viewing rectangle by . How does the value of affect the graph?

Question 37

Graphs of the functions f and g are given.
(a) Which is larger,
(b) Which is larger,
(c) For which values of x is

Question 38

A function is given.
(a) Find all the local maximum and minimum values of the function and the value of x at which each occurs.
(b) Find the intervals on which the function is increasing and on which the function is decreasing. State all answers correct to two decimal places.

Question 39

Sketch the graph of the piecewise-defined function.

Question 40

The graph of a function h is given.
(a) Find
(b) Find the domain and range of h.
(c) Find the values of x for which
(d) Find the values of x for which

Question 41

The graph of a function is given. Determine the average rate of change of the function between the indicated points on the graph.

Question 42

For the function determine the average rate of change between the values and .

Question 43

A man is running around a circular track that is 200 m in circumference. An observer uses a stopwatch to record the runner's time at the end of each lap, obtaining the data in the following table.
What was the man's average speed (rate) between 108 s and 203 s? Round the answer to two decimal places.

Question 44

For the function determine the average rate of change between the values and .

Question 45

For the function determine the average rate of change between the values and .

Question 46

A man is running around a circular track that is 200 m in circumference. An observer uses a stopwatch to record the runner's time at the end of each lap, obtaining the data in the following table.
What was the man's average speed (rate) between 68 s and 203 s? Round the answer to two decimal places.

Question 47

The graph of a function is given. Determine the average rate of change of the function between the indicated points on the graph.

Question 48

For the function determine the average rate of change between the values and

Question 49

For the function determine the average rate of change between the values and .

Question 50

The graph shows the depth of water W in a reservoir over a one-year period as a function of the number of days x since the beginning of the year. What was the average rate of change of W between x = 0 and x= 100?

Question 51

For the function determine the average rate of change between the values and .

Question 52

For the function determine the average rate of change between the values and ( ).

Question 53

Given the graph of $f$ , describe how the graph of $y = - f ( 2 x ) + 2$ can be obtained from the graph of $f$

A) Stretch horizontally by a factor of $2$ , then reflecting about the
$y$ - axis, then shifting
$2$ units up
B) Shrink vertically by a factor of $2$ , then reflecting about the
$x$ - axis, then shifting
$2$ units up.
C) Stretch horizontally by a factor of $2$ , then reflecting about the
$y$ - axis, then shifting
$2$ units up .
D) Shrink vertically by a factor of $\frac { 1 } { 2 }$ , then reflecting about the
$x$ - axis, then shifting downward
$2$ units.
E) Shrink horizontally by a factor of $\frac { 1 } { 2 }$ , then reflecting about the
$x$ - axis, then shifting
$2$ units up

Question 54

For the function determine the average rate of change between the values and

Question 55

The graph shows the depth of water W in a reservoir over a one-year period as a function of the number of days x since the beginning of the year. Estimate the average rate of change of W between x = 200 and x= 300.

Question 56

Given the graph of $f ,$ describe how the graph of $y = \frac { 4 } { 3 } f ( x - 3 ) + \frac { 4 } { 3 }$ can be obtained from the graph of $f$

A) Shift $\frac { 4 } { 3 }$ units to the right, then stretching vertically by a factor of
$3$ , then shift upward
$\frac { 4 } { 3 }$ units.
B) Shift $3$ units to the right, then shrinking horizontally by a factor of
$\frac { 4 } { 3 }$ , then shift downward
$\frac { 4 } { 3 }$ units.
C) Shift $3$ units to the left, then shrinking vertically by a factor of
$\frac { 4 } { 3 }$ , then shift upward
$3$ units.
D) Shift $3$ units to the right, then stretching vertically by a factor of
$\frac { 4 } { 3 }$ , then shift upward
$\frac { 4 } { 3 }$ units.
E) Shift $3$ units to the left, then shrinking vertically by a factor of
$3$ , then shift upward
$\frac { 4 } { 3 }$ units.

Question 57

The graph shows the depth of water W in a reservoir over a one-year period as a function of the number of days x since the beginning of the year. What was the average rate of change of W between x = 100 and x= 200?

Question 58

A function is given.
(a) Find all the local maximum and minimum values of the function and the value of x at which each occurs.
(b) Find the intervals on which the function is increasing and on which the function is decreasing. State all answers correct to two decimal places.

Question 59

A function is given.
(a) Find all the local maximum and minimum values of the function and the value of x at which each occurs.
(b) Find the intervals on which the function is increasing and on which the function is decreasing. State all answers correct to two decimal places.

Question 60

For the linear function , find the average rate of change between and . Show that the average rate of change is the same as the slope of the line.

Question 61

Let and . Find , and their domains.

Question 62

The graph of is shown. Use this graph to sketch the graph of .

Question 63

Determine whether is even or odd. If is even or odd, use symmetry to sketch its graph.

Question 64

Let and . Find , , and their domains.

Question 65

Given , , and , find .

Question 66

Graph the given functions on the same screen using the viewing rectangle by . How is each part related to the graph in part
(a)

(a)?
(b)

(c)

(d)

Question 67

Given and , find , , , and , and their domains.

Question 68

Use $f ( x ) = x ^ { 2 } - 1$ and $g ( x ) = - \sqrt { x + 1 }$ to evaluate the expression $( f \circ g ) ( x )$ .

A) $- 1$
B) $- x ^ { 2 }$
C) $x ^ { 2 }$
D) $x$
E) $0$

Question 69

Sketch the graph of the function , not by plotting points, but by starting with the graph of a standard function and applying transformations.

Question 70

For the functions and , find , , , and their domains. 10.
Given and , find .

Question 71

If and , find , , and their domains.

Question 72

Use $f ( x ) = 3 x - 2$ and $g ( x ) = 3 + 2 x ^ { 2 }$ to evaluate the expression $( f \circ g ) ( 2 )$ .

A) $31$
B) $14$
C) $39$
D) $11$
E) $114$  

Question 73

Explain how the graph of $g ( x ) = \frac { | x + 1 | - 2 } { 3 }$ is obtained from the graph of $f = | x |$ .

A) Shift the graph of $f$ one unit to the right, then shift downward
$\frac { 2 } { 3 }$ units.
B) Shift the graph of $f$ one unit to the right, then shift downward
$2$ units.
C) Shift the graph of $f$ one unit to the left, then shrink vertically by a factor of
$\frac { 2 } { 3 }$ .
D) Shift the graph of $f$ one unit to the left, then shift upward
$\frac { 2 } { 3 }$ units.
E) Shift the graph of $f$ one unit to the left, shrink vertically by a factor of
$\frac { 1 } { 3 }$ , shift downward
$\frac { 2 } { 3 }$ units.

Question 74

Use $f ( x ) = x - 2$ and $g ( x ) = 4 - 2 x ^ { 2 }$ to evaluate the expression $g ( f ( 0 ) )$ .

A) $- 3$
B) $- 2$
C) $- 4$
D) $0$
E) $1$

Question 75

Shift the graph of $f ( x ) = x ^ { 2 }$ by $6$ units to the left and then $4$ units downward, and choose the equation for the final transformed graph.

A) $f ( x ) = \left( x ^ { 2 } + 6 \right) - 4$
B) $f ( x ) = ( x + 6 ) ^ { 2 } - 4$
C) $f ( x ) = ( x + 4 ) ^ { 2 } + 6$
D) $f ( x ) = ( x - 6 ) ^ { 2 } - 4$
E) $f ( x ) = ( x - 4 ) ^ { 2 } + 6$

Question 76

Use $f ( x ) = x - 3$ and $g ( x ) = 3 + x ^ { 2 }$ to evaluate the expression $( g \circ f ) ( - 1 )$ .

A) $14$
B) $19$
C) $3$
D) $12$
E) $- 12$

Question 77

Use $f ( x ) = x - 4$ and $g ( x ) = 2 - x ^ { 2 }$ to evaluate the expression $g ( g ( 1 ) )$ .

A) $1$
B) $3$
C) $- 1$
D) $- 3$
E) $0$

Question 78

If and , find , , and their domains.

Question 79

Explain how the graph of $g ( x ) = - \sqrt [ 3 ] { x + 1 }$ is obtained from the graph of $f = \sqrt [ 3 ] { x }$

A) Shift the graph of $f$ one unit to the left, then reflect about the
$y$ - axis.
B) Shift the graph of $f$ one unit to the left, then reflect about the
$y$ - axis .
C) Shift the graph of $f$ one unit up, reflect about the
$x$ - axis and stretch vertically by a factor of
$\frac { 1 } { 3 }$ .
D) Shift the graph of $f$ one unit to the left, then reflect about the
$x$ - axis.
E) Shift the graph of $f$ one unit down, reflect about the
$x$ - axis and stretch vertically by a factor of
$\frac { 1 } { 3 } \text {. }$

Question 80

If and , find , , and their domains.