Deck 15: Functions and Precalculus

Question 1

Which of the following points lie on the graph of $$f ( x ) = \sqrt { 3 x + 4 }$$ ?A (-3, 6) B (4, 4) C (0, 4)

A) A and B
B) B and C
C) Only B
D) Only C

Answer 1

Only B

Question 2

Describe the function $f ( t ) = - 5 t + 1$ as a set of ordered pairs.

Answer 2

$( t , 5 t + 1 )$

Question 3

Construct a rule $f : \{ 1,2,3,4,5 \} \rightarrow \{ 3,6,9,12 \}$ that is (a) a function,(b) a one-to-one function.

Answer 3

(a) $$f ( 1 ) = f ( 2 ) = 3 , f ( 3 ) = 6 , f ( 4 ) = 9 , \text { and } f ( 5 ) = 12$$ (b) $$f ( 1 ) = 3 , f ( 3 ) = 6 , f ( 4 ) = 9 , \text { and } f ( 5 ) = 12$$
(There are other possible answers.)

Question 4

Find the domain of $f ( x ) = \frac { \sqrt { x ^ { 2 } - 4 } } { x ^ { 2 } - 9 }$

Question 5

Find the average rate of change of the function $f ( x ) = \sqrt { x + 1 }$ on the interval [0, 8].

A) 8/3
B) 3/8
C) 1/4
D) -1/4

Question 6

Find the domain and range of $f ( x ) = \sqrt { 2 - x } + 1$

Question 7

Find the domain and range of $f ( x ) = \frac { 1 } { \sqrt { x - 3 } }$

A) Domain: $( - \infty , 3 ]$ Range: $[ 0 , \infty )$
B) Domain: $[ 0 , \infty )$ Range: $( - \infty , 3 ]$
C) Domain: $( - \infty , 3 )$ Range: $( 0 , \infty )$
D) Domain: $( 3 , \infty )$ Range: $( 0 , \infty )$

Question 8

Find the domain and range of $f ( x ) = \frac { 1 } { x ^ { 2 } + 4 }$

A) Domain: $( - \infty , - 2 ) \cup ( - 2,2 ) \cup ( 2 , \infty )$ Range: $( - \infty , \infty )$
B) Domain: $( - \infty , \infty )$ Range: $( - \infty , \infty )$
C) Domain: $( - \infty , \infty )$ Range: $( 0 , \infty )$
D) Domain: $( - \infty , - 2 ) \cup ( - 2,2 ) \cup ( 2 , \infty )$ Range: $[ 0 , \infty )$

Question 9

Find the domain and range of $f ( x ) = \frac { | x - 2 | } { x - 2 }$

A) Domain: $( - \infty , - 2 ) \cup ( - 2,2 ) \cup ( 2 , \infty )$ Range: $( - \infty , \infty )$
B) Domain: $( - \infty , \infty )$ Range: $( - \infty , \infty )$
C) Domain: $( - \infty , - 2 ) \cup ( - 2,2 ) \cup ( 2 , \infty )$ Range: $( - \infty , \infty )$
D) Domain: $( - \infty , - 2 ) \cup ( - 2,2 ) \cup ( 2 , \infty )$ Range: $\{ - 1,1 \}$

Question 10

Find the domain and range of $f ( x ) = \frac { \sqrt { 9 - x ^ { 2 } } } { x + 3 }$

A) Domain: [-3, 3] Range: $[ 0 , \infty )$
B) Domain: (-3, 3) Range: $[ 0 , \infty )$
C) Domain: (-3, 3] Range: $[ 0 , \infty )$
D) Domain: (-3, 3] Range: $( 0 , \infty )$

Question 11

Find the domain and range of $f ( x ) = \sqrt { x ^ { 2 } - 5 x + 4 }$

A) Domain: $( - \infty , - 4 ) \cup ( - 1 , \infty )$ Range: $( 0 , \infty )$
B) Domain: $( - \infty , 1 ) \cup ( 4 , \infty )$ Range: $[ 0 , \infty )$
C) Domain: $( - \infty , 1 ] \cup [ 4 , \infty )$ Range: $[ 0 , \infty )$
D) Domain: $( - \infty , 1 ] \cup [ 4 , \infty )$ Range: $( 0 , \infty )$

Question 12

If $f ( x ) = x ^ { 3 } + 2$ , find $f \left( a ^ { 2 } \right)$

A) $a ^ { 5 } + 2$
B) $\left( a ^ { 2 } + 2 \right) ^ { 3 }$
C) $a ^ { 3 } + 8$
D) $a ^ { 6 } + 2$

Question 13

Given $y = 1 - \sqrt { x }$ , for what value(s) of $X$ is $y = - 2$ ?

A) 4
B) 9
C) 16
D) None of these

Question 14

Find the domain of $f ( x ) = \frac { x ^ { 2 } - 9 } { \sqrt { x + 3 } }$

A) Domain: $[ - 3 , \infty )$
B) Domain: $( - 3 , \infty )$
C) Domain: $( - \infty , \infty )$
D) Domain: $( - \infty , - 3 ) \cup ( - 3 , \infty )$

Question 15

Given $y = x ^ { 2 } - 4 x - 8$ , for what value(s) of $X$ is $y = - 3$ ?

A) -5 and 1
B) 5 and 1
C) 4 and 2
D) -1 and 5

Question 16

If $f ( x ) = \sqrt { 2 x - 1 }$ , find $\frac { f ( 1 + h ) - f ( 1 ) } { h }$

Question 17

If $g ( x ) = \frac { x ^ { 2 } + 1 } { x }$ , find $g ( x + h ) - g ( x )$

Question 18

Which of the following is an odd function?

A) $f ( x ) = x ^ { 3 } - x ^ { 2 }$
B) $g ( x ) = \frac { 5 } { x }$
C) $h ( x ) = x ^ { 4 } - x ^ { 2 }$
D) $k ( x ) = | x - 2 |$

Question 19

Is $f ( x ) = \frac { 2 x ^ { 3 } } { x ^ { 2 } + 2 }$ an even function, an odd function or neither? Explain.

Question 20

Is $g ( x ) = \frac { - 3 } { x ^ { 2 } + 1 }$ an even function, an odd function or neither? Explain.

Question 21

Is $g ( x ) = \frac { x ^ { 5 } - x } { x ^ { 2 } + 1 }$ an even function, an odd function or neither? Explain.

Question 22

If $f$ is an odd function and $g$ is an even function, determine whether the following functions are odd, even or neither.
a. $( f ( x ) ) ^ { 2 } + ( g ( x ) ) ^ { 3 }$
b. $5 + g ( x )$
c. $f \left( x ^ { 3 } \right)$
d. $f ( g ( x ) )$

Question 23

(-3, 4) is on the graph of an odd function $f ( x )$ find another point on the graph of $f ( x )$

A) (-3, 3)
B) (-3, 4)
C) (3, 4)
D) (3, -4)

Question 24

(2, 3) is on the graph of an even function $g ( x )$ find another point on the graph of $g ( x )$

A) (2, -3)
B) (-2, -3)
C) (-2, 3)
D) (-3, 2)

Question 25

Complete the entries in the table below to make $f$ an even function
$$\begin{array} { | l | l | l | l | l | l | l | l | } \hline x & - 3 & - 2 & - 1 & 0 & 1 & 2 & 3 \\\hline f ( x ) & 5 & & - 2 & & & 3 & \\\hline\end{array}$$

Question 26

Complete the entries in the table below to make $f$ an odd function
$$\begin{array} { | l | l | l | l | l | l | l | l | } \hline x & - 3 & - 2 & - 1 & 0 & 1 & 2 & 3 \\\hline f ( x ) & 5 & & - 2 & & & 3 & \\\hline\end{array}$$

Question 27

Find the inverse of $f ( x ) = \frac { 1 + 4 x } { 3 }$

A) $f ^ { - 1 } ( x ) = \frac { 3 x + 1 } { 4 }$
B) $f ^ { - 1 } ( x ) = \frac { 4 x - 1 } { 3 }$
C) $f ^ { - 1 } ( x ) = \frac { 4 x + 1 } { 3 }$
D) $f ^ { - 1 } ( x ) = \frac { 3 x - 1 } { 4 }$

Question 28

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

A) $f ^ { - 1 } ( x ) = \frac { x - 2 } { x + 2 }$
B) $f ^ { - 1 } ( x ) = \frac { 2 x - 1 } { x + 2 }$
C) $f ^ { - 1 } ( x ) = \frac { 2 x - 2 } { x - 1 }$
D) $f ^ { - 1 } ( x ) = \frac { 2 x + 2 } { x - 1 }$

Question 29

Find the inverse of $f ( x ) = - \sqrt { 2 - 5 x }$

Question 30

Let $f ( x ) = 2 x ^ { 3 } + 3 x + 1$ Find $X$ if $f ^ { - 1 } ( x ) = - 1$

A) 3/2
B) -4
C) 0
D) 4

Question 31

Let $f ( x ) = \frac { x ^ { 2 } } { 1 + x ^ { 3 } }$ Find $X$ if $f ^ { - 1 } ( x ) = 2$

A) $1 / 2$
B) 9/4
C) 4/9
D) 2/5

Question 32

Given that $f ( x ) = x ^ { 2 } + 2$ , and $g ( x ) = \sqrt { x }$ , find $( f \circ g ) ( x )$ and $( g \circ f ) ( x )$ , and their domains.

Question 33

Given that $f ( x ) = \frac { 1 } { x } + 1$ and $g ( x ) = \frac { 1 } { 1 + x }$ , find $( f \circ g ) ( 2 )$ , and their domains.

Question 34

Table below defines two functions $f$ and $g$ Create an additional row for the table for the function $( g \circ f ) ( x )$
$$\begin{array} { | l | l | l | l | l | l | l | l | } \hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\\hline f ( x ) & 0 & 1 & 3 & 2 & 3 & 0 & 2 \\\hline g ( x ) & 1 & 0 & 1 & 1 & 0 & 1 & 0 \\\hline\end{array}$$

Question 35

Table below defines two functions $f$ and $g$ Create an additional row for the table for the function $( f \circ g ) ( x )$
$$\begin{array} { | l | l | l | l | l | l | l | l | } \hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\\hline f ( x ) & 0 & 1 & 3 & 2 & 3 & 0 & 2 \\\hline g ( x ) & 1 & 0 & 1 & 1 & 0 & 1 & 0 \\\hline\end{array}$$

Question 36

Table below defines two functions $f$ and $g$ Create an additional row for the table for the function $g ( x + 1 )$
$$\begin{array} { | l | l | l | l | l | l | l | l | } \hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\\hline f ( x ) & 0 & 1 & 3 & 2 & 3 & 0 & 2 \\\hline g ( x ) & 1 & 0 & 1 & 1 & 0 & 1 & 0 \\\hline\end{array}$$

Question 37

Express $f ( x ) = \left( 2 + \cos \left( x ^ { 3 } \right) \right) ^ { 2 }$ as a composition of two functions: $g$ and $h$ such that $f = g \circ h$

Question 38

Express $f ( x ) = \sqrt [ 3 ] { 1 + \sqrt { x } }$ as a composition of three functions: $g$ and $h$ such that $f = g \circ h$

Question 39

Given $f ( x ) = x ^ { 2 } + 1$ and $g ( x ) = 3 x + 2$ , find all values of $X$ such that $f ( g ( x ) ) = g ( f ( x ) )$

A) 2, 3
B) 0, 2
C) 0, - 2
D) 0, 3

Question 40

Given $f ( x ) = 2 x + 1$ and $h ( x ) = 2 x ^ { 2 } + 4 x + 1$ , find a function $g$ such that $f ( g ( x ) ) = h ( x )$

A) $g ( x ) = \left( x ^ { 2 } + 1 \right) ^ { 2 }$
B) $g ( x ) = x ^ { 2 } + 2$
C) $g ( x ) = x ^ { 2 } + 4 x$
D) $g ( x ) = x ^ { 2 } + 2 x$

Question 41

Given that $f ( x ) = x ^ { 2 } - 1$ , $g ( x ) = \frac { 1 } { x + 3 }$ , and $h ( x ) = \sqrt { x }$ , find an equation for the following functions, evaluate them at x = 1 and find their domains.
(a) $( 4 f h ) ( x )$
(b) $\left( \frac { f } { g } \right) ( x )$

Question 42

Given that $f ( x ) = x ^ { 2 } - 1$ , $g ( x ) = \frac { 1 } { x + 3 }$ , and $h ( x ) = \sqrt { x }$ , find an equation for $f ( x - 2 ) + 3 h ( x ) + ( 2 f g ) ( x )$ , evaluate it at x = 1 and find its domain.

Question 43

Write all the transformations applied to obtain the graph of $y = - x ^ { 2 } + 4 x$ from the graph of $y = x ^ { 2 }$

Question 44

Write all the transformations applied to obtain the graph of $y = | 2 x + 4 | - 3$ from the graph of $y = | x |$

Question 45

Write all the transformations applied to obtain the graph of $y = 2 - \sqrt { x + 1 }$ from the graph of $y = \sqrt { x }$

Question 46

Suppose (3, 7) is a point on the graph of $f ( x )$ , find a point on the graph of $f ( x ) - 3$

A) (0, 7)
B) (3, 4)
C) (3, 7)
D) (0, 4)

Question 47

Suppose (3, 7) is a point on the graph of $f ( x )$ , find a point on the graph of $f ( x - 2 )$

A) (-2, 7)
B) (3, 5)
C) (1, 7)
D) (1,5)

Question 48

Suppose (3, 7) is a point on the graph of $f ( x )$ , find a point on the graph of $2 f ( x + 1 ) - 3$

A) (4, 7)
B) (4, 14)
C) (4, 11)
D) (8, 11)

Question 49

Given $g ( x ) = \left\{ \begin{array} { c c } 3 & x < 1 \\\sqrt { x + 1 } & x \geq 1\end{array} \right.$ Find $g \left( t ^ { 2 } - 1 \right)$

Question 50

Find the domain and range of the piecewise defined function: $$f ( x ) = \left\{ \begin{array} { c c } - 1 & x \leq - 5 \\\frac { 1 } { x } & - 5 < x < 5 \\x - 5 & x > 5\end{array} \right.$$

A) Domain: $( - \infty , \infty )$ Range: $( - \infty , \infty )$
B) Domain: $( - \infty , 5 ) \cup ( 5 , \infty )$ Range: $( - \infty , \infty )$
C) Domain: $( - \infty , 0 ) \cup ( 0,5 ) \cup ( 5 , \infty )$ Range: $[ 0 , \infty )$
D) Domain: $( - \infty , 0 ) \cup ( 0,5 ) \cup ( 5 , \infty )$ Range: $\{ - 1 \} \cup [ 0 , \infty )$

Question 51

$f ( x ) = \left\{ \begin{array} { r r } x ^ { 2 } + 3 & x < 0 \\3 - x & x \geq 0\end{array} \right.$ Calculate $f ( - 1 ) , f ( 0 )$ and $f ( 2 )$

Question 52

A. If $f$ is an invertible function such that $f ( 3 ) = 3$ , then $f ^ { - 1 } ( 3 ) = \frac { 1 } { 3 }$ .
B. A one-to-one function is invertible.
C. If $f$ and $g$ are inverses of each other then $f$ and $g$ have the same domain.

A)A and B are true.
B)B and C are true.
C) Only B is true.
D) None of these

Question 53

Write $f ( x ) = | 3 - 2 x |$ as a piecewise function.

A) $f ( x ) = \left\{ \begin{array} { c c } - 3 + 2 x & x \leq 3 / 2 \\3 - 2 x & x > 3 / 2\end{array} \right.$
B) $f ( x ) = \left\{ \begin{array} { c c } - 3 - 2 x & x \leq 3 / 2 \\3 + 2 x & x > 3 / 2\end{array} \right.$
C) $f ( x ) = \left\{ \begin{array} { c c } - 3 + 2 x & x < 3 / 2 \\3 - 2 x & x \geq 3 / 2\end{array} \right.$
D) $f ( x ) = \left\{ \begin{array} { c c } - 3 + 2 x & x > 3 / 2 \\3 - 2 x & x \leq 3 / 2\end{array} \right.$

Question 54

Write $f ( x ) = \left| 4 - x ^ { 2 } \right|$ as a piecewise function.

A) $f ( x ) = \left\{ \begin{array} { l } 4 + x ^ { 2 } - 2 \leq x \leq 2 \\4 - x ^ { 2 } \text { otherwise }\end{array} \right.$
B) $f ( x ) = \left\{ \begin{array} { c c } - 4 + x ^ { 2 } & - 2 \leq x \leq 2 \\4 - x ^ { 2 } & \text { otherwise }\end{array} \right.$
C) $f ( x ) = \left\{ \begin{array} { c c } 4 + x ^ { 2 } & - 2 \leq x \leq 2 \\- 4 + x ^ { 2 } & \text { otherwise }\end{array} \right.$
D) $f ( x ) = \left\{ \begin{array} { c c } 4 - x ^ { 2 } & - 2 \leq x \leq 2 \\- 4 + x ^ { 2 } & \text { otherwise }\end{array} \right.$

Question 55

Write $f ( x ) = 5 + | 3 x - 2 |$ as a piecewise function.

A) $f ( x ) = \left\{ \begin{array} { c c } - 3 + 3 x & x < 2 / 3 \\3 - 3 x & x \geq 3 / 2\end{array} \right.$
B) $f ( x ) = \left\{ \begin{array} { c c } - 3 - 3 x & x < 2 / 3 \\3 + 3 x & x \geq 2 / 3\end{array} \right.$
C) $f ( x ) = \left\{ \begin{array} { l l } 7 - 3 x & x < 2 / 3 \\3 + 3 x & x \geq 2 / 3\end{array} \right.$
D) $f ( x ) = \left\{ \begin{array} { l l } 7 + 3 x & x < 2 / 3 \\3 - 3 x & x \geq 2 / 3\end{array} \right.$

Question 56

Find the equation of a linear function whose graph has slope -2 and passes through the point (-1, 2). Write your answer in slope-intercept form.

A) $y = - 2 x - 5$
B) $y = - 2 x + 3$
C) $y = - 2 x + 4$
D) $y = - 2 x$

Question 57

Find the equation of a linear function whose graph passes through the points (2, -1) and (-3, 2).

A) $5 x + 3 y + 7 = 0$
B) $5 x + 3 y + 3 = 0$
C) $3 x + 5 y + 3 = 0$
D) $3 x + 5 y + 7 = 0$

Question 58

Find the equation of a linear function whose graph is perpendicular to $x - 2 y + 2 = 0$ and passes through the point (1, 3).

A) $y = - 2 x - 5$
B) $y = - 2 x + 1$
C) $y = - 2 x + 5$
D) $y = 2 x - 5$

Question 59

Given $f ( x ) = \frac { x ^ { 2 } + x - 2 } { x - 1 }$ , find a linear function $g$ which is identical to $f$

Question 60

Sketch the graph of a possible rational function $f$ with roots at $x = - 1$ and $x = 3$ , a vertical asymptote at $x = 2$ , and a horizontal asymptote at $y = - 2$

Question 61

Construct an equation of a rational function whose graph has no roots, no holes, has vertical asymptotes at $x = - 3 , x = 3$ and a horizontal asymptote at $y = 2$ There are many possible answers, but choose the only correct answer from those that are provided below.

A) $f ( x ) = \frac { 2 x ^ { 2 } - 1 } { x ^ { 2 } - 9 }$
B) $f ( x ) = \frac { 2 x ^ { 2 } - 1 } { x ^ { 2 } + 9 }$
C) $f ( x ) = \frac { 2 x ^ { 2 } + 1 } { x ^ { 2 } - 9 }$
D) $f ( x ) = \frac { 2 x ^ { 2 } + 1 } { x ^ { 2 } - 3 }$

Question 62

Find an equation for a power function whose graph passes through the points (0, 0) and (2, 5).

Question 63

Find an equation for a polynomial function whose graph passes through the points (3, 0), (-1, 0) and (-3, 0).

A) $f ( x ) = x ^ { 3 } - x ^ { 2 } - 9 x - 9$
B) $f ( x ) = x ^ { 3 } - x ^ { 2 } + 9 x + 9$
C) $f ( x ) = x ^ { 3 } - x ^ { 2 } + 9 x - 9$
D) $f ( x ) = x ^ { 3 } + x ^ { 2 } - 9 x - 9$

Question 64

For the polynomial $f ( x ) = x ^ { 3 } ( 3 - x ) ^ { 2 } \left( 1 + x ^ { 2 } \right)$ , determine the leading coefficient and the degree.

A) $x ^ { 7 } , 3$
B) $x ^ { 7 } , 1$
C) $- x ^ { 7 } , 3$
D) $- x ^ { 7 } , 1$

Question 65

Explain why the graph of $f ( x ) = \frac { ( x - 2 ) ( x + 3 ) } { ( x + 3 ) ^ { 2 } }$ does not have a hole in it at $x = - 3$ ?

Question 66

Find the domain of $f ( x ) = \frac { \ln ( x + 1 ) } { \ln ( x - 3 ) }$

A) $( - 1 , \infty )$
B) $( 3 , \infty )$
C) $( - 1,3 ) \cup ( 3 , \infty )$
D) $( 3,4 ) \cup ( 4 , \infty )$

Question 67

Find the domain of $f ( x ) = \frac { \ln ( x - 3 ) } { \ln ( x + 1 ) }$

A) (-1, $\infty$ )
B) $( - 1,0 ) \cup ( 0 , \infty )$
C) (3, $\infty$ )
D) [3, $\infty$ )

Question 68

Find the domain of $f ( x ) = \frac { 1 } { e ^ { 3 x } - e ^ { x } }$

Question 69

Find the exact value of $6 \log _ { 2 } 6 - 3 \log _ { 2 } 9$

Question 70

Solve the following equation for $X$ : $\log _ { 2 } \frac { x + 1 } { x - 2 } = 3$

Question 71

Solve the following equation for $X$ : $5 e ^ { - 2 x } = 6$

Question 72

Solve the following equation for $X$ : $e ^ { x } - 4 x e ^ { x } = 0$

A) 0
B) 0 and $1 / 4$
C) $1 / 4$
D) 4

Question 73

Solve the following equation for $X$ : $e ^ { 2 x } - e ^ { x } - 2 = 0$

A) 0
B) 2
C) ln(2)
D) -1

Question 74

Solve the following equation for $X$ : $\ln ( 6 x ) - 3 \ln \left( x ^ { 2 } \right) = \ln ( 3 )$

A) $2 ^ { 5 }$
B) 2
C) ln(2)
D) $\sqrt [ 5 ] { 2 }$

Question 75

Find the domain and range of $f ( x ) = 1 + \ln ( x - 3 )$

Question 76

Find the domain and range of $f ( x ) = 2 + e ^ { x - 1 }$

Question 77

Expand the logarithm in terms of sums and differences: $$\ln \frac { x ^ { 3 } \cos ^ { 2 } x } { \sqrt { x ^ { 2 } + 1 } }$$

Question 78

Is $f ( x ) = \sin ^ { 2 } x$ an even function, odd function or neither?

Question 79

Express the following as a polynomial function of $X :$ $5 \exp ( \ln 3 ) + 2 \ln \left( e ^ { 3 x } \left( e ^ { x } \right) ^ { 2 } \right)$

Question 80

Which of the following expressions is/are defined?
A. $\sin ^ { - 1 } \left( - \frac { 1 } { 5 } \right)$
B. $\sin ^ { - 1 } \left( \frac { 3 } { 2 } \right)$
C. $\tan ^ { - 1 } ( 2 )$

A) A and B
B) B and C
C) A and C
D) Only A