Deck 3: Polynomial and Rational Functions

Question 1

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

Answer 1

Domain:
, Range

Question 2

Sketch the graph of the function.

Answer 2

Question 3

Answer 3

Question 4

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 each answer correct to two decimal places.

Question 5

Use a graphing device to draw the graph of the function $f ( x ) = 144 x ^ { 3 } - 144 x ^ { 2 } + 36 x$
State approximately the intervals on which the function is increasing and on which the function is decreasing.

Question 6

Determine whether the function in the figure is even, odd, or neither.

Question 7

Describe how the graph of
can be obtained from the graph of

Question 8

For the function $f ( x ) = 2 x ^ { 2 } - x$
determine the average rate of change between the values $x = - 1$
and $x = 0$

Question 9

$$\text { If } f ( x ) = 2 x ^ { 2 } + 1 \text { and } g ( x ) = x - 1 \text {, find } f + g , f g \text {, and their domains. }$$

Question 10

Find the range of the function. $f ( x ) = - x ^ { 2 } - 2 x + 3$

Question 11

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 12

Determine whether or not the function $f ( x ) = x ^ { 2 } - 3 x + 2$
is one-to-one.

Question 13

Given
and
, find
,
.

Question 14

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

Question 15

Find the domain of the function. $f ( x ) = 0.5 x - \frac { 2 } { \sqrt { x + 1 } }$

Question 16

Sketch the graph of the function.

Question 17

Question 18

If an object is dropped from a high cliff or a tall building, then the distance it has fallen after t seconds is given by the function
Find its average speed (average rate of change) over the following intervals:
(i) Between 1 s and 6 s
(ii) Between

Question 19

Draw the graph of the function in an appropriate viewing rectangle.

Question 20

Question 21

Use a graphing calculator to estimate the range of the function.

Question 22

Find the domain of the function. $f ( x ) = \frac { x + 3 } { x ^ { 2 } - 4 }$

Question 23

Sketch the graph of the function.

Question 24

Sketch the graph of the piecewise defined function.

Question 25

Use a graphing calculator to determine approximately the intervals on which the function is increasing, and on which $f$ is decreasing. $f ( x ) = x ^ { 4 } + 6 x ^ { 3 } + x ^ { 2 } - 24 x + 16$

Question 26

Determine whether each curve represents a graph of a function.
a.)

b.)

c.)

d.)

Question 27

If an object is dropped from a high cliff or a tall building, then the distance it has fallen after t seconds is given by the function $f ( t ) = 16 t ^ { 2 }$
Find its average speed (average rate of change) over the following intervals:
(i) Between 2 s and 7 s
(ii) Between $t = c \text { and } t = c + h$

Question 28

For the function $g ( t ) = \frac { 1 } { 3 t - 2 }$
determine the average rate of change between the values $t = 0$ and $t = a + 1$ .

Question 29

Determine if the equation $x ^ { 2 } + y ^ { 2 } - 25 = 0$
defines $y$ as a function of $x$ . Explain your answer.

Question 30

A function is given. Use a graphing calculator to draw the graph of f. Find the domain and range of f from the graph. $f ( x ) = - \sqrt { 16 - x ^ { 2 } }$

Question 31

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 each answer correct to two decimal places.

Question 32

Describe how the graph of $y = - f ( 3 x ) + 4$
can be obtained from the graph of $f .$

Question 33

A function f is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. $f ( x ) = | x |$ ; shift to the left $1 / 2$ unit, shrink vertically by a factor of 0.2, and shift downward 2 units.

Question 34

Find the inverse of the function. $f ( x ) = \sqrt { 25 - x ^ { 2 } }$ , $0 \leq x \leq 5$

Question 35

Sketch the graph of the function
.

Question 36

Determine whether $f ( x ) = x ^ { 2 } - x ^ { 6 }$
is even, odd, or neither.

Question 37

Use a graphing calculator or computer to determine whether or not the function $f ( x ) = 2 x ^ { 3 } - x$ is one-to-one.

Question 38

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 39

Find the inverse of the function. $f ( x ) = \frac { 1 } { 2 } x + 1$

Question 40

For the function given, find $g ( - 1 )$ , $g ( 3 )$ , and $g \left( a ^ { 2 } \right)$

$g ( x ) = ( 1 / x ) + x ^ { 2 }$

Question 41

Determine which viewing rectangle produces the most appropriate graph of the function.

Question 42

Use a graphing device to draw the graph of the function $f ( x ) = - 3 - 3 x ^ { 2 }$
. State approximately the intervals on which the function is increasing and on which the function is decreasing.

Question 43

Find the inverse of the function. $g ( x ) = x ^ { 2 } - 9 , x \geq 0$

Question 44

Determine whether the curve represents a graph of a function.

A) Function; the graph passes the horizontal line test
B) Function; the graph passes the vertical line test
C) Not a function; the graph passes the vertical line test
D) Not a function; the graph passes the horizontal line test
E) none

Question 45

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 = 0 and x= 100?

Question 46

Determine whether or not the function
is one-to-one.

Question 47

Use a graphing calculator to determine approximately the internals on which the function $f$ is decreasing. $f ( x ) = x ^ { 4 } + 6 x ^ { 3 } + x ^ { 2 } - 24 x + 16$

A) $[ - 4 , - 1.5 ] , [ 1 , \infty )$
B) $( - \infty , - 4 ] , [ - 1.5,1 )$
C) $( 0 , \infty )$
D) $( 0 , - 16 )$
E) $( - \infty , - 16 ]$

Question 48

Use a graphing calculator to find, approximately the range of the function.

Question 49

If an object is dropped from a high cliff or a tall building, then the distance it has fallen after t seconds is given by the function Find its average speed (average rate of change) over the interval between

A)
B)
C)
D)
E) none of these

Question 50

Find the inverse of the function. $f ( x ) = 3 x + 2$

Question 51

Given $f ( x ) = 2 + x ^ { 2 }$ and $g ( x ) = \sqrt { x - 1 }$
, find $( f \circ g ) ( 2 )$ , $( f \circ f ) ( 2 )$

Question 52

Evaluate $f ( - 1 )$ , $f ( 0 )$ , $f ( 1 )$
, for the piecewise-defined function. $$f ( x ) = \left\{ \begin{array} { l l } x ^ { 2 } & \text { if } x < 0 \\2 x - 1 & \text { if } x \geq 0\end{array} \right.$$

A) $f ( - 1 ) = 1 , f ( 0 ) = - 1 , f ( 1 ) = 1$
B) $f ( - 1 ) = - 3 , f ( 0 ) = 0 , f ( 1 ) = 1$
C) $f ( - 1 ) = - 1 , f ( 0 ) = 0 , f ( 1 ) = 1$
D) $f ( - 1 ) = 3 , f ( 0 ) = 1 , f ( 1 ) = - 3$
E) none of these

Question 53

Sketch the graph of the function e)

Question 54

Determine whether the function in the figure is even, odd, or neither.

Question 55

For the function $f ( t ) = \frac { 1 } { 3 t - 2 }$ determine the average rate of change between the values $t = 0$
And $t = b + 1$

A) $\frac { 3 } { ( 3 b - 1 ) }$

B) $\frac { 2 } { 3 ( b + 1 ) }$

C) $3 ( 3 b + 1 )$

D) $\frac { - 1 } { ( 2 b - 3 ) }$

E) $\frac { 3 } { 2 ( 3 b + 1 ) }$

Question 56

Use a graphing calculator or computer to determine whether or not the function $f ( x ) = - | x | - | 5 - x |$ is one-to-one.

Question 57

A function f is given, and the indicated transformations are applied to its graph (in the given order). Find the equation for the final transformed graph.

Question 58

Sketch the graph of the function.

Question 59

If $f ( x ) = 3 x - 2$ and $g ( x ) = 3 + 2 x ^ { 2 }$
, find $f g$ and $( f \circ g ) ( x )$

Question 60

Determine if the equation $x ^ { 2 } + y ^ { 2 } = 49$ defines y as a function of x

A) The equation represents a function because it's a circle
B) The equation represents a function because for each value of x there is always two values of y
C) Not a function because the equation gives two values of y for a given value of x
D) The equation represents a function because it passes the vertical line test
E) Not a function because the equation passes the vertical line test.

Question 61

Find the domain of the function.

Question 62

For the function $f ( t ) = \frac { 1 } { t }$ determine the average rate of change between the values $t = \tau$ And $t = a + h$

A) $\frac { - 1 } { a ( a + h ) }$

B) $\frac { - 2 } { h }$

C) $( 3 h + 1 )$

D) $\frac { - 1 } { ( 2 h - 1 ) }$

E) $\frac { 1 } { a ( a + h ) }$

Question 63

Evaluate $f ( - 1 )$ , $f ( 0 )$ , $f ( 1 )$
, for the piecewise-defined function. $f ( x ) = \left\{ \begin{array} { l l } x ^ { 2 } & \text { if } x < 0 \\ x - 1 & \text { if } x \geq 0 \end{array} \right.$

A) $f ( - 1 ) = 1 , f ( 0 ) = - 1 , f ( 1 ) = 0$
B) $f ( - 1 ) = - 2 , f ( 0 ) = 0 , f ( 1 ) = 1$
C) $f ( - 1 ) = - 1 , f ( 0 ) = 0 , f ( 1 ) = 1$
D) $f ( - 1 ) = 2 , f ( 0 ) = 1 , f ( 1 ) = - 3$
E) none of these

Question 64

Find the range of the function.

Question 65

Find the inverse of the function.

A)
B)
C)
D)
E) none of these

Question 66

Use a graphing calculator to find, approximately the range of the function.

A)
B)
C)
D)
E)

Question 67

Determine whether the curve represents a graph of a function.

A) Function; the graph passes the horizontal line test
B) Function; the graph passes the vertical line test
C) Not a function; the graph passes the vertical line test
D) Not a function; the graph passes the horizontal line test
E) none

Question 68

Determine which functions are one-to-one.

A) I only
B) I, II
C) I, II, III
D) II only
E) III only

Question 69

Find the inverse of the function.

Question 70

Determine which viewing rectangle produces the most appropriate graph of the function.

A)
B)
C)
D)
E)

Question 71

Question 72

Given $f ( x ) = 2 + x ^ { 2 }$ and $g ( x ) = \sqrt { x - 4 }$ , find $( f \circ g ) ( 4 )$

A) $( f \circ g ) ( 4 ) = 2$
B) $( f \circ g ) ( 4 ) = 4$
C) $( f \circ g ) ( 4 ) = - 2$
D) $( f \circ g ) ( 4 ) = 0$
E) $( f \circ g ) ( 4 ) = 1$

Question 73

If $f ( x ) = 3 x - 2$ and $g ( x ) = 3 + 2 x ^ { 2 }$ , find $f g$ and $( g \circ f ) ( x )$

A) $f g = x ^ { 3 } - 4 x ^ { 2 } + 9 x - 1 ; \quad \quad ( g \circ f ) ( x ) = 18 x ^ { 2 } - 24 x + 11$
B) $f g = 6 x ^ { 3 } - 4 x ^ { 2 } + 9 x - 6 ; \quad (g \circ f ) ( x ) = 18 x ^ { 2 } - 24 x + 11$
C) $f g = 2 x ^ { 2 } + 3 x + 1$ ; $\quad$ $\quad$ $\quad$$( g \circ f ) ( x ) = 6 x ^ { 2 } + 7$
D) $f g = 6 x ^ { 2 } + 7$ ; $\quad$ $\quad$ $\quad$ $\quad$$( g \circ f ) ( x ) = 6 x ^ { 3 } - 4 x ^ { 2 } + 9 x - 6$
E) none of these

Question 74

Find the average rate of change of the function between the points given.

A)
B)
C)
D)
E)

Question 75

Sketch the graph of the function. $$f ( x ) = \left\{ \begin{array} { l l c } - x & \text { if } \quad x < - 1 \\x ^ { 2 } & \text { if } & - 1 \leq x \leq 1 \\1 & \text { if } & x > 1\end{array} \right.$$

A)

B)

C)

D)

E) none

Question 76

Determine if the equation defines y as a function of x

A) The equation represents a function because it's a circle
B) The equation represents a function because for each value of x there is always two values of y
C) Not a function because the equation gives two values of y for a given value of x
D) The equation represents a function because it passes the vertical line test
E) Not a function because the equation passes the vertical line test.

Question 77

Question 78

Question 79

A function f is given, and the indicated transformations are applied to its graph (in the given order). Find the equation for the final transformed graph. ; shift 5 units to the left, stretch vertically by a factor of 2, and reflect in the x-axis

A)
B)
C)
D)
E)

Question 80

Sketch the graph of the function.