Question 1

Express the function $F ( x ) = 3 \sqrt { x } + 2$ in the form $f \circ g$ .

Accepted Answer

The answer of Express the function \(F ( x )...

Question 2

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

Accepted Answer

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 
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 3

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

Accepted Answer

The average rate of

Question 4

Suppose that the total cost $C$ (in thousands of dollars) to produce $x$ units (in millions) of a particular product is given by the function $C ( x ) = \frac { x ^ { 2 } } { 10 } + 100$ . 
(a) Find
   . 
(b) Find
   . What does your answer represent?

Accepted Answer

(a) Let @#LAT-DLM& @#LAT-DLM& @#LAT-DLM& @#LAT-DLM& @#LAT-DLM& @#LAT-DLM& @#LAT-DLM& , but @#LAT-DLM& , so @#LAT-DLM& @#LAT-DLM& @#LAT-DLM&

Question 5

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.

Accepted Answer

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)  $f ( x ) = ( x - 4 ) ^ { 2 } + 6$ 
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 6

Determine if the equation $2 x ^ { 2 } y - y - 5 = 0$ defines $y$ as a function of $x$ . Explain your answer.

Accepted Answer

@#LAT-DLM& @#LAT-DLM& @#LAT-DLM& @#LAT-DLM& @#LAT-DLM& or @#LAT-DLM& . No. Since this equa

Question 7

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

Accepted Answer

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
$\frac { 1 } { 3 } \text {. }$ 
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 8

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

Accepted Answer

@#LAT-DLM& , with domain @#LAT-DLM& @#LAT-DLM& , w

Question 9

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 $f$

Accepted Answer

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. 
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 10

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

Accepted Answer

Average ra

Question 11

For the function $f ( x ) = 2 x - 2$ determine the average rate of change between the values $x = 5$ and $x = 6$ .

Accepted Answer

Average ra

Question 12

Let $f ( x ) = c x ^ { 4 }$ . Graph the family of functions with $c = \frac { 1 } { 4 }$ , $\frac { 1 } { 2 }$ , $1$ , $2$ , and $4$ in the viewing rectangle $[ - 2,2 ]$ by $[ - 1,8 ]$ . How does the value of $c$ affect the graph?

Accepted Answer

For @#LAT-DLM& , the

Question 13

For the function $f ( t ) = 2 t ^ { 2 } - t$ determine the average rate of change between the values $t = 2$ and $t = 2 + h$ ( $h \neq 0$ ).

Accepted Answer

Average ra

Question 14

The graph of a function h is given.
(a) Find $h ( - 3 ) , h ( - 2 ) , h ( 0 ) , \text { and } h ( 3 )$
(b) Find the domain and range of h.
(c) Find the values of x for which $h ( x ) = 3$
(d) Find the values of x for which $h ( x ) \leq 3$

Accepted Answer

(a) @#LAT-DLM&
(b) @#LAT-DLM&

Question 15

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

Accepted Answer

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
 $[ 6 , - 4 ]$ ,
 $[ - 4,0 ]$ 
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 16

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?

Accepted Answer

The answer of The graph shows the depth of water...

Question 17

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

Accepted Answer

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 
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 18

A function is given. Use a graphing calculator to draw the graph of f. Find the domain and range of f from the graph. $f ( x ) = 5 , \quad 1 \leq x \leq 3$

Accepted Answer

The answer of A function is given. Use a graphing...

Question 19

Graph the given functions on the same screen using the viewing rectangle $[ - 5,7 ]$ by $[ - 10,10 ]$ . How is each part related to the graph in part 
(a)
 $y = x ^ { 2 }$ 
(a)?
(b)
 $y = \frac { 1 } { 2 } x ^ { 2 }$ 
(c)
 $y = \frac { 1 } { 2 } ( x - 3 ) ^ { 2 }$ 
(d)
 $y = - \frac { 1 } { 2 } ( x - 3 ) ^ { 2 } - 3$

Accepted Answer

b) Shrunk vertically by a factor of @#LAT-DLM& . (

Question 20

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.
 $G ( x ) = \frac { 3 } { x ^ { 2 } + x + 1 }$

Accepted Answer

a) local maximum @#LAT-DLM& wh

Express the function $F ( x ) = 3 \sqrt { x } + 2$ in the form $f \circ g$ .

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

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

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.

Determine if the equation $2 x ^ { 2 } y - y - 5 = 0$ defines $y$ as a function of $x$ . Explain your answer.

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

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

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 $f$

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

For the function $f ( x ) = 2 x - 2$ determine the average rate of change between the values $x = 5$ and $x = 6$ .

Let $f ( x ) = c x ^ { 4 }$ . Graph the family of functions with $c = \frac { 1 } { 4 }$ , $\frac { 1 } { 2 }$ , $1$ , $2$ , and $4$ in the viewing rectangle $[ - 2,2 ]$ by $[ - 1,8 ]$ . How does the value of $c$ affect the graph?

For the function $f ( t ) = 2 t ^ { 2 } - t$ determine the average rate of change between the values $t = 2$ and $t = 2 + h$ ( $h \neq 0$ ).

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

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?

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 is given. Use a graphing calculator to draw the graph of f. Find the domain and range of f from the graph. $f ( x ) = 5 , \quad 1 \leq x \leq 3$

Fundamentals

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions: Unit Circle Approach

Trigonometric Functions: Right Triangle Approach

Analytic Trigonometry

Polar Coordinates and Parametric Equations

Vectors in Two and Three Dimensions

Systems of Equations and Inequalities

Conic Sections

Sequences and Series

Limits: a Preview of Calculus

Filters

Exam 2: Functions

Express the function F(x)=3x+2F ( x ) = 3 \sqrt { x } + 2F(x)=3x​+2 in the form f∘gf \circ gf∘g .

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

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

Suppose that the total cost CCC (in thousands of dollars) to produce xxx units (in millions) of a particular product is given by the function C(x)=x210+100C ( x ) = \frac { x ^ { 2 } } { 10 } + 100C(x)=10x2​+100 . (a) Find . (b) Find . What does your answer represent?

Shift the graph of f(x)=x2f ( x ) = x ^ { 2 }f(x)=x2 by 666 units to the left and then 444 units downward, and choose the equation for the final transformed graph.

Determine if the equation 2x2y−y−5=02 x ^ { 2 } y - y - 5 = 02x2y−y−5=0 defines yyy as a function of xxx . Explain your answer.

Explain how the graph of g(x)=−x+13g ( x ) = - \sqrt [ 3 ] { x + 1 }g(x)=−3x+1​ is obtained from the graph of f=x3f = \sqrt [ 3 ] { x }f=3x​

Given f(x)=1x+2f ( x ) = \frac { 1 } { x + 2 }f(x)=x+21​ and g(x)=1x−2g ( x ) = \frac { 1 } { x - 2 }g(x)=x−21​ , find f∘gf \circ gf∘g , g∘fg \circ fg∘f , f∘ff \circ ff∘f , and g∘gg \circ gg∘g , and their domains.

Given the graph of f,f ,f, describe how the graph of y=43f(x−3)+43y = \frac { 4 } { 3 } f ( x - 3 ) + \frac { 4 } { 3 }y=34​f(x−3)+34​ can be obtained from the graph of fff

For the function g(t)=13t−2g ( t ) = \frac { 1 } { 3 t - 2 }g(t)=3t−21​ determine the average rate of change between the values t=0t = 0t=0 and t=a+1t = a + 1t=a+1 .

For the function f(x)=2x−2f ( x ) = 2 x - 2f(x)=2x−2 determine the average rate of change between the values x=5x = 5x=5 and x=6x = 6x=6 .

Let f(x)=cx4f ( x ) = c x ^ { 4 }f(x)=cx4 . Graph the family of functions with c=14c = \frac { 1 } { 4 }c=41​ , 12\frac { 1 } { 2 }21​ , 111 , 222 , and 444 in the viewing rectangle [−2,2][ - 2,2 ][−2,2] by [−1,8][ - 1,8 ][−1,8] . How does the value of ccc affect the graph?

For the function f(t)=2t2−tf ( t ) = 2 t ^ { 2 } - tf(t)=2t2−t determine the average rate of change between the values t=2t = 2t=2 and t=2+ht = 2 + ht=2+h ( h≠0h \neq 0h=0 ).

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

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?

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

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)=5,1≤x≤3f ( x ) = 5 , \quad 1 \leq x \leq 3f(x)=5,1≤x≤3

Fundamentals

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions: Unit Circle Approach

Trigonometric Functions: Right Triangle Approach

Analytic Trigonometry

Polar Coordinates and Parametric Equations

Vectors in Two and Three Dimensions

Systems of Equations and Inequalities

Conic Sections

Sequences and Series

Limits: a Preview of Calculus

Filters

Express the function $F ( x ) = 3 \sqrt { x } + 2$ in the form $f \circ g$ .

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.

Determine if the equation $2 x ^ { 2 } y - y - 5 = 0$ defines $y$ as a function of $x$ . Explain your answer.

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

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

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 $f$

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

For the function $f ( x ) = 2 x - 2$ determine the average rate of change between the values $x = 5$ and $x = 6$ .

Let $f ( x ) = c x ^ { 4 }$ . Graph the family of functions with $c = \frac { 1 } { 4 }$ , $\frac { 1 } { 2 }$ , $1$ , $2$ , and $4$ in the viewing rectangle $[ - 2,2 ]$ by $[ - 1,8 ]$ . How does the value of $c$ affect the graph?

For the function $f ( t ) = 2 t ^ { 2 } - t$ determine the average rate of change between the values $t = 2$ and $t = 2 + h$ ( $h \neq 0$ ).

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 is given. Use a graphing calculator to draw the graph of f. Find the domain and range of f from the graph. $f ( x ) = 5 , \quad 1 \leq x \leq 3$