Question 1

Let $f ( x ) = - \frac { | x | } { x ^ { 2 } }$ .
(a) Sketch the graph of $f$ .
(b) Find the domain and the range of $f$ .

Accepted Answer

(a) @#IMG-DLM&
(b) D

Question 2

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

Accepted Answer

@#LAT-DLM& , domain

Question 3

Find the inverse function of $f ( x ) = 2 - \frac { 1 } { x }$ , $x \neq 0$ .

Accepted Answer

The answer of Find the inverse function of \(f (...

Question 4

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: $( - \infty , - 2 ) \cup ( - 2 , \infty )$ , 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 
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 5

Given $f ( x ) = \frac { 1 } { x }$ , $g ( x ) = \frac { x } { x + 1 }$ , and $h ( x ) = \frac { x + 1 } { x }$ , find $f \circ g \circ h$ .

Accepted Answer

The answer of Given \(f ( x ) = \frac...

Question 6

If $f ( x ) = 2 x ^ { 2 } + x$ and $g ( x ) = 3 x - 1$ , find $f + g$ , $f - g$ , and their domains.

Accepted Answer

@#LAT-DLM& , domain

Question 7

Let $f ( x ) = 2 x - 1$ .
(a) Sketch the graph of $f$ .
(b) Find the domain of $f$ .
(c) State the intervals on which $f$ is increasing and on which $f$ is decreasing.

Accepted Answer

(a) @#IMG-DLM&
(b) D

Question 8

Determine whether or not the function $g ( x ) = 1 - x ^ { 2 } , x \geq 0$ is one-to-one.

Accepted Answer

@#LAT-DLM&
Two different positive numbers cannot h

Question 9

For the functions $f ( x ) = \frac { 1 } { x - 2 }$ and $g ( x ) = \frac { x } { x - 1 }$ , find $f + g$ , $f - g$ , $\mathrm { fg }$ , $\frac { f } { g }$ and their domains. Answer $f + g = \frac { 1 } { x - 2 } + \frac { x } { x - 1 } = \frac { x ^ { 2 } - x - 1 } { ( x - 2 ) ( x - 1 ) } \quad$ domain: $( - \infty , 1 ) \cup ( 1,2 ) \cup ( 2 , \infty )$
$$\begin{array} { l l } 
f - g = \frac { 1 } { x - 2 } - \frac { x } { x - 1 } = - \frac { x ^ { 2 } - 3 x + 1 } { ( x - 2 ) ( x - 1 ) } & \text { domain: } ( - \infty , 1 ) \cup ( 1,2 ) \cup ( 2 , \infty ) \
f g = \left( \frac { 1 } { x - 2 } \right) \left( \frac { x } { x - 1 } \right) = \frac { x } { ( x - 1 ) ( x - 2 ) } & \text { domain: } ( - \infty , 1 ) \cup ( 1,2 ) \cup ( 2 , \infty ) \
\frac { f } { g } = \frac { ( x - 1 ) } { ( x - 2 ) x } & \text { domain: } ( - \infty , 0 ) \cup ( 0,1 ) \cup ( 1,2 ) \cup ( 2 , \infty )
\end{array}$$ 10.
Given $f ( x ) = 2 x - 2$ and $g ( x ) = 2 x ^ { 2 }$ , find $( g \circ f ) ( - 1 )$ .

Accepted Answer

The answer of For the functions \(f ( x )...

Question 10

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.
$$\begin{array} { | c | c | } 
\hline \text { Time (s) } & \begin{array} { c } 
\text { Distance } \
( \mathrm { m } )
\end{array} \
\hline 32 & 200 \
\hline 68 & 400 \
\hline 108 & 600 \
\hline 152 & 800 \
\hline 203 & 1000 \
\hline 263 & 1200 \
\hline 335 & 1400 \
\hline 412 & 1600 \
\hline
\end{array}$$

Accepted Answer

The answer of A man is running around a circular...

Question 11

Graphs of the functions f and g are given.
(a) Which is larger, $f ( - 3 ) \text { or } g ( 3 ) ?$
(b) Which is larger, $f ( - 1 ) \text { or } g ( - 1 ) ?$
(c) For which values of x is $f ( x ) = g ( x ) ?$

Accepted Answer

The answer of Graphs of the functions f and g...

Question 12

A function is given. Use a graphing calculator to draw the graph of f. Find the domain and range of f from the graph. $f ( x ) = x ^ { 2 } , \quad - 3 \leq x \leq 5$

Accepted Answer

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

Question 13

Find the inverse function of $f ( x ) = 3 ( x + 1 ) ^ { 2 }$ , $x < - 1$ .

Accepted Answer

@#LAT-DLM& @#LAT-DLM& @#LAT-DLM& @#LAT-DLM& @#LAT-DLM& , but

Question 14

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

Accepted Answer

@#LAT-DLM&
is not one-to-one, or sketch

Question 15

For the function $f ( x ) = 2 x ^ { 2 } - 1$ , find $f ( x + 1 )$ and $f ( x ) + 1$ .

Accepted Answer

The answer of For the function \(f ( x )...

Question 16

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

Accepted Answer

@#LAT-DLM& , domain

Question 17

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

Accepted Answer

(a) local maximum @#LAT-DLM& w

Question 18

Let $f ( x ) = 3 x + 1$ and $g ( x ) = 3 x ^ { 2 } - 2 x + 1$ . Find $f - g$ , $\mathrm { fg }$ , and their domains.

Accepted Answer

@#LAT-DLM& , domain

Question 19

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

Accepted Answer

The answer of Sketch the graph of the piecewise-defined function....

Question 20

Let $f ( x ) = 3 + x ^ { 2 }$ , $0 \leq x \leq 5$ 
(a) Sketch the graph of
 $f$ then use it to sketch the graph of
 $f ^ { - 1 }$ . 
(b) Find
 $f ^ { - 1 }$ .

Accepted Answer

@#LAT-DLM& @#LAT-DLM& @#LAT-DLM& . So th

Let $f ( x ) = - \frac { | x | } { x ^ { 2 } }$ . (a) Sketch the graph of $f$ . (b) Find the domain and the range of $f$ .

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

Find the inverse function of $f ( x ) = 2 - \frac { 1 } { x }$ , $x \neq 0$ .

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

Given $f ( x ) = \frac { 1 } { x }$ , $g ( x ) = \frac { x } { x + 1 }$ , and $h ( x ) = \frac { x + 1 } { x }$ , find $f \circ g \circ h$ .

If $f ( x ) = 2 x ^ { 2 } + x$ and $g ( x ) = 3 x - 1$ , find $f + g$ , $f - g$ , and their domains.

Let $f ( x ) = 2 x - 1$ . (a) Sketch the graph of $f$ . (b) Find the domain of $f$ . (c) State the intervals on which $f$ is increasing and on which $f$ is decreasing.

Determine whether or not the function $g ( x ) = 1 - x ^ { 2 } , x \geq 0$ is one-to-one.

A function is given. Use a graphing calculator to draw the graph of f. Find the domain and range of f from the graph. $f ( x ) = x ^ { 2 } , \quad - 3 \leq x \leq 5$

Find the inverse function of $f ( x ) = 3 ( x + 1 ) ^ { 2 }$ , $x < - 1$ .

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

For the function $f ( x ) = 2 x ^ { 2 } - 1$ , find $f ( x + 1 )$ and $f ( x ) + 1$ .

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

Let $f ( x ) = 3 x + 1$ and $g ( x ) = 3 x ^ { 2 } - 2 x + 1$ . Find $f - g$ , $\mathrm { fg }$ , and their domains.

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

Let $f ( x ) = 3 + x ^ { 2 }$ , $0 \leq x \leq 5$ (a) Sketch the graph of $f$ then use it to sketch the graph of $f ^ { - 1 }$ . (b) Find $f ^ { - 1 }$ .

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

Let f(x)=−∣x∣x2f ( x ) = - \frac { | x | } { x ^ { 2 } }f(x)=−x2∣x∣​ . (a) Sketch the graph of fff . (b) Find the domain and the range of fff .

If f(x)=2x2+1f ( x ) = 2 x ^ { 2 } + 1f(x)=2x2+1 and g(x)=x−1g ( x ) = x - 1g(x)=x−1 , find f+gf + gf+g , fg\mathrm { fg }fg , and their domains.

Find the inverse function of f(x)=2−1xf ( x ) = 2 - \frac { 1 } { x }f(x)=2−x1​ , x≠0x \neq 0x=0 .

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

Given f(x)=1xf ( x ) = \frac { 1 } { x }f(x)=x1​ , g(x)=xx+1g ( x ) = \frac { x } { x + 1 }g(x)=x+1x​ , and h(x)=x+1xh ( x ) = \frac { x + 1 } { x }h(x)=xx+1​ , find f∘g∘hf \circ g \circ hf∘g∘h .

If f(x)=2x2+xf ( x ) = 2 x ^ { 2 } + xf(x)=2x2+x and g(x)=3x−1g ( x ) = 3 x - 1g(x)=3x−1 , find f+gf + gf+g , f−gf - gf−g , and their domains.

Let f(x)=2x−1f ( x ) = 2 x - 1f(x)=2x−1 . (a) Sketch the graph of fff . (b) Find the domain of fff . (c) State the intervals on which fff is increasing and on which fff is decreasing.

Determine whether or not the function g(x)=1−x2,x≥0g ( x ) = 1 - x ^ { 2 } , x \geq 0g(x)=1−x2,x≥0 is one-to-one.

Graphs of the functions f and g are given. (a) Which is larger, f(−3) or g(3)?f ( - 3 ) \text { or } g ( 3 ) ?f(−3) or g(3)? (b) Which is larger, f(−1) or g(−1)?f ( - 1 ) \text { or } g ( - 1 ) ?f(−1) or g(−1)? (c) For which values of x is f(x)=g(x)?f ( x ) = g ( x ) ?f(x)=g(x)?

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)=x2,−3≤x≤5f ( x ) = x ^ { 2 } , \quad - 3 \leq x \leq 5f(x)=x2,−3≤x≤5

Find the inverse function of f(x)=3(x+1)2f ( x ) = 3 ( x + 1 ) ^ { 2 }f(x)=3(x+1)2 , x<−1x < - 1x<−1 .

Determine whether or not the function h(x)=x3−xh ( x ) = x ^ { 3 } - xh(x)=x3−x is one-to-one.

For the function f(x)=2x2−1f ( x ) = 2 x ^ { 2 } - 1f(x)=2x2−1 , find f(x+1)f ( x + 1 )f(x+1) and f(x)+1f ( x ) + 1f(x)+1 .

If f(x)=2−2xf ( x ) = \sqrt { 2 - 2 x }f(x)=2−2x​ and g(x)=x2−1g ( x ) = \sqrt { x ^ { 2 } - 1 }g(x)=x2−1​ , find f+gf + gf+g , fg\mathrm { fg }fg , and their domains.

Let f(x)=3x+1f ( x ) = 3 x + 1f(x)=3x+1 and g(x)=3x2−2x+1g ( x ) = 3 x ^ { 2 } - 2 x + 1g(x)=3x2−2x+1 . Find f−gf - gf−g , fg\mathrm { fg }fg , and their domains.

Sketch the graph of the piecewise-defined function. f(x)={x2 if ∣x∣≤1−x2 if ∣x∣>1f ( x ) = \left\{ \begin{array} { l l } x ^ { 2 } & \text { if } | x | \leq 1 \\- x ^ { 2 } & \text { if } | x | > 1\end{array} \right.f(x)={x2−x2​ if ∣x∣≤1 if ∣x∣>1​

Let f(x)=3+x2f ( x ) = 3 + x ^ { 2 }f(x)=3+x2 , 0≤x≤50 \leq x \leq 50≤x≤5 (a) Sketch the graph of fff then use it to sketch the graph of f−1f ^ { - 1 }f−1 . (b) Find f−1f ^ { - 1 }f−1 .

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

Let $f ( x ) = - \frac { | x | } { x ^ { 2 } }$ . (a) Sketch the graph of $f$ . (b) Find the domain and the range of $f$ .

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

Find the inverse function of $f ( x ) = 2 - \frac { 1 } { x }$ , $x \neq 0$ .

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

Given $f ( x ) = \frac { 1 } { x }$ , $g ( x ) = \frac { x } { x + 1 }$ , and $h ( x ) = \frac { x + 1 } { x }$ , find $f \circ g \circ h$ .

If $f ( x ) = 2 x ^ { 2 } + x$ and $g ( x ) = 3 x - 1$ , find $f + g$ , $f - g$ , and their domains.

Let $f ( x ) = 2 x - 1$ . (a) Sketch the graph of $f$ . (b) Find the domain of $f$ . (c) State the intervals on which $f$ is increasing and on which $f$ is decreasing.

Determine whether or not the function $g ( x ) = 1 - x ^ { 2 } , x \geq 0$ is one-to-one.

A function is given. Use a graphing calculator to draw the graph of f. Find the domain and range of f from the graph. $f ( x ) = x ^ { 2 } , \quad - 3 \leq x \leq 5$

Find the inverse function of $f ( x ) = 3 ( x + 1 ) ^ { 2 }$ , $x < - 1$ .

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

For the function $f ( x ) = 2 x ^ { 2 } - 1$ , find $f ( x + 1 )$ and $f ( x ) + 1$ .

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

Let $f ( x ) = 3 x + 1$ and $g ( x ) = 3 x ^ { 2 } - 2 x + 1$ . Find $f - g$ , $\mathrm { fg }$ , and their domains.

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

Let $f ( x ) = 3 + x ^ { 2 }$ , $0 \leq x \leq 5$ (a) Sketch the graph of $f$ then use it to sketch the graph of $f ^ { - 1 }$ . (b) Find $f ^ { - 1 }$ .