Deck 3: Functions

Question 1

Find the domain. $g ( t ) = \frac { 1 } { t ^ { 2 } - 7 t + 12 }$

A) $( - \infty , - 4 ) \cup ( - 4,3 ) \cup ( 3 , \infty )$
B) $( - \infty , 3 ) \cup ( 3,4 ) \cup ( 4 , \infty )$
C) $( - \infty , - 4 ) \cup ( 3 , \infty )$
D) $( - \infty , 4 ) \cup ( 5 , \infty )$
E) $( - \infty , - 3 ) \cup ( 3 , \infty )$

Answer 1

$( - \infty , 3 ) \cup ( 3,4 ) \cup ( 4 , \infty )$

Question 2

Determine the domain and the range. $F ( x ) = \frac { x ^ { 3 } + 6 } { x ^ { 3 } - 2 }$

A) domain: $( - \infty , \sqrt [ 3 ] { 2 } ) \cup ( \sqrt [ 3 ] { 2 } , \infty )$ ; range: $( - \infty , \infty )$
B) domain: $( - \infty , \sqrt [ 3 ] { 2 } ) \cup ( \sqrt [ 3 ] { 2 } , \infty )$ ; range: $( - \infty , 1 ) \cup ( 1 , \infty )$
C) domain: $\left( - \infty , 2 ^ { 3 } \right) \cup \left( 2 ^ { 3 } , \infty \right)$ ; range: $( - \infty , 1 ) \cup ( 1 , \infty )$
D) domain: $( - \infty , - \sqrt [ 3 ] { 2 } ) \cup ( - \sqrt [ 3 ] { 2 } , \infty )$ ; range: $( - \infty , \infty )$
E) domain: $( - \infty , \sqrt [ 3 ] { 2 } ) \cup ( \sqrt [ 3 ] { 2 } , \infty )$ ; range: $( - \infty , 0 ) \cup ( 0 , \infty )$

Answer 2

domain: $( - \infty , \sqrt [ 3 ] { 2 } ) \cup ( \sqrt [ 3 ] { 2 } , \infty )$ ; range: $( - \infty , 1 ) \cup ( 1 , \infty )$

Question 3

Let T(x) = 2x ² - 3x. Find (and simplify) the expression. T (x + 3) - T (x - 3)

A) 24x
B) 18x - 24
C) 24x - 18
D) 6x
E) 18x

Answer 3

24x - 18

Question 4

Find the domain. $g ( t ) = \sqrt { t ^ { 2 } - 5 t + 6 }$

A) $( - \infty , - 3 ] \cup [ 2 , \infty )$
B) $( - \infty , 2 ) \cup ( 2,3 ) \cup ( 3 , \infty )$
C) $[ 2,3 ]$
D) $( - \infty , 2 ] \cup [ 3 , \infty )$
E) $( - \infty , 2 ) \cup ( 3 , \infty )$

Question 5

Using the results in the table, graph the functions y = x ² , y = x ² - 2, and y = x ² + 2 on the same set of axes. x
X ²
X ² - 2
X ² + 2
0
0
-2
2 $\pm$ 1
1
-1
3 $\pm$ 2
4
2
6 $\pm$ 3
9
7
11

A)
B)
C)
D)
E)

Question 6

Given that f and g are inverse functions. If f(7) = 12, what is g(12)?

A) 17
B) 12
C) 5
D) 4
E) 7

Question 7

Find the domain. $g ( t ) = \sqrt [ 3 ] { t ^ { 2 } - 9 t + 20 }$

A) $( - \infty , \infty )$
B) $( - \infty , 4 ) \cup ( 5 , \infty )$
C) $[ 4,5 ]$
D) $( - \infty , 4 ) \cup ( 4,5 ) \cup ( 5 , \infty )$
E) $[ - 4 , \infty )$

Question 8

Find the domain. $h ( x ) = \sqrt { x ^ { 2 } - 49 }$

A) $( 7 , \infty )$
B) $( - \infty , 49 ] \cup [ 49 , \infty )$
C) $( - \infty , - 7 ) \cup ( 7 , \infty )$
D) $( - \infty , - 7 ] \cup [ 7 , \infty )$
E) $( - 7,7 )$

Question 9

Specify the domain and range of the given function. (The axes are marked in one-unit intervals.)

A) domain: $[ - 2,4 )$ ; range: $[ - 3,3 ]$
B) domain: $[ - 3,1 ) \cup ( 1,3 ]$ ; range: $[ - 2,4 ]$
C) domain: $[ - 3,3 ]$ ; range: $[ - 2,4 )$
D) domain: $[ - 3,3 ]$ ; range: $[ - 2 , - 1 )$
E) domain: $[ - 3,1 ) \cup ( 1,3 ]$ ; range: $[ - 2,4 )$

Question 10

A spherical weather balloon is being inflated in such a way that the radius is given by $r = g ( t ) = \frac { 1 } { 2 } t + 2$ Assume that r is in meters and t is in seconds, with t = 0 corresponding to the time that inflation begins. If the volume of a sphere of radius r is given by $V ( r ) = \frac { 4 } { 3 } \pi r ^ { 3 }$ find the time at which the volume of the balloon is $36 \pi \mathrm { m } ^ { 3 }$

A) t = 5 sec
B) t = 6 sec
C) t = 3 sec
D) t = 2 sec
E) t = 1 sec

Question 11

Find the domain. $H ( t ) = \sqrt [ 3 ] { \frac { 5 t - 4 } { 7 - 2 t } }$

A) $\left( - \infty , \frac { 4 } { 5 } \right) \cup \left( \frac { 4 } { 5 } , \frac { 7 } { 2 } \right) \cup \left( \frac { 7 } { 2 } , \infty \right)$
B) $\left( - \infty , \frac { 4 } { 5 } \right] \cup \left( \frac { 7 } { 2 } , \infty \right)$
C) $\left( \frac { 4 } { 5 } , \frac { 7 } { 2 } \right)$
D) $\left( - \infty , \frac { 7 } { 2 } \right) \cup \left( \frac { 7 } { 2 } , \infty \right)$
E) $\left( - \infty , \frac { 4 } { 5 } \right) \cup \left( \frac { 7 } { 2 } , \infty \right)$

Question 12

Sketch the graph. $$y = \left\{ \begin{array} { l l } 2 \sqrt { 1 - x ^ { 2 } } & \text { if } - 1 \leq x < 1 \\\frac { 2 } { x } & \text { if } x \geq 1\end{array} \right.$$

A)
B)
C)
D)
E)

Question 13

Let f (x) = x ² . Find and simplify the indicated difference quotient. $\frac { f ( y + h ) - f ( y ) } { h }$

A) y ² - h
B) 2y - h
C) 2y + h ²
D) 2y + h
E) y + h

Question 14

Set up a table and graph the function $y = x ^ { 2 } - 2$ .

A)
B)
C)
D)
E)

Question 15

Let f(x) = x ³ + 2x + 1, and assume that the domain of f ^-1 is $( - \infty , \infty )$ . Simplify $f \left[ f ^ { - 1 } ( t + 5 ) \right]$ .

A) t ³ + 3t + 5
B) (t + 5) ²
C) $\frac { 1 } { t + 5 }$
D) t ² + 2t + 1
E) t + 5

Question 16

Set up a table and graph the function $y = 1 / x .$

A)
B)
C)
D)
E)

Question 17

Let T (x) = 4x ² - 3x. Find (and simplify) the expression. T (x + h) - T (x - h)

A) 16h - 3x
B) 16hx - 6h
C) 8x + 4h
D) 8x - 3h
E) 16hx + 3h

Question 18

Suppose that in a certain biology lab experiment, the number of bacteria is related to the temperature T of the environment by the function $N ( T ) = - 2 T ^ { 2 } + 240 T - 5,100 ( 40 \leq t \leq 90 )$ Here, N(T) represents the number of bacteria present when the temperature is T degrees Fahrenheit. Also, suppose that t hr after the experiment begins, the temperature is given by $T ( t ) = 10 t + 40 ( 0 \leq t \leq 5 )$ How many bacteria are present when t = 5 hr?

A) 300
B) 900
C) 1,200
D) 400
E) 600

Question 19

Determine the domain and the range. $g ( x ) = \frac { 11 x - 20 } { 3 x - 18 }$

A) domain: $\left( - \infty , \frac { 11 } { 3 } \right) \cup \left( \frac { 11 } { 3 } , \infty \right)$ range: $( - \infty , 6 ) \cup ( 6 , \infty )$ ;
B) domain: $( - \infty , 6 ) \cup ( 6 , \infty )$ ; range: $\left( - \infty , \frac { 20 } { 11 } \right) \cup \left( \frac { 20 } { 11 } , \infty \right)$
C) domain: $( - \infty , 6 ) \cup ( 6 , \infty )$ ; range: $\left( - \infty , \frac { 11 } { 3 } \right) \cup \left( \frac { 11 } { 3 } , \infty \right)$
D) domain: $( - \infty , 6 ) \cup ( 6 , \infty )$ ; range: $\left( - \infty , \frac { 3 } { 11 } \right) \cup \left( \frac { 3 } { 11 } , \infty \right)$
E) domain: $( - \infty , 6 ) \cup ( 6 , \infty )$ ; range: $\left( - \infty , \frac { 11 } { 18 } \right) \cup \left( \frac { 11 } { 18 } , \infty \right)$

Question 20

Assume that the accompanying viewing rectangle shows the essential features of the graph of f(x) = -x ³ + 4x+ 2. Use a graphing utility to estimate to the nearest hundredth the coordinates of the turning points. f(x) = -x ³ + 4x+ 2 [- 4, 4, 2] by [- 12, 12, 6]

A) (- 1.14, - 1.08),(1.14, 5.08)
B) (- 1.14, - 1.09),(1.14, 5.09)
C) (- 1.15, - 1.07),(1.15, 5.07)
D) (- 1.15, - 1.08),(1.15, 5.08)
E) (- 1.16, - 1.08),(1.16, 5.08)

Question 21

Let $g ( t ) = \frac { 4 } { t } + 6$ . Find g ^{- 1} (t).

A) $g ^ { - 1 } ( t ) = \frac { 4 } { t - 6 }$
B) $g ^ { - 1 } ( t ) = \frac { 4 } { t } + 6$
C) $g ^ { - 1 } ( t ) = \frac { t } { 4 } - 6$
D) $g ^ { - 1 } ( t ) = \frac { 7 } { t } - 6$
E) $g ^ { - 1 } ( t ) = \frac { 6 } { t - 4 }$

Question 22

Use a graphing utility to graph the function and then apply the horizontal line test to see whether the function is one-to-one. y = x ⁷ + 5x

A) True
B) False

Question 23

Let f(x) = 5x + 4. Find f ^{- 1} (x).

A) $f ^ { - 1 } ( x ) = \frac { x + 5 } { 4 }$
B) $f ^ { - 1 } ( x ) = \frac { x - 4 } { 5 }$
C) $f ^ { - 1 } ( x ) = \frac { x + 4 } { 8 }$
D) $f ^ { - 1 } ( x ) = \frac { x } { 5 } - 4$
E) $f ^ { - 1 } ( x ) = \frac { 1 } { 5 x - 4 }$

Question 24

Assume that the domain of $f$ and $f ^ { - 1 }$ is $( - \infty , \infty )$ . Solve for t. $$f \left( \frac { 6 - 2 t } { 6 + 2 t } \right) = 12 ; f ^ { - 1 } ( 12 ) = - 3$$ .

A) t = - 6
B) t = 12
C) t = 7
D) t = - 7
E) t = -7