Question 1

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

Accepted Answer

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

Question 2

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.

Accepted Answer

$$\begin{array} { l l } 
f ( g ( x ) ) = x + 2 & \text { Domain: } [ 0 , \infty ) \
g ( f ( x ) ) = \sqrt { x ^ { 2 } + 2 } & \text { Domain: } ( - \infty , \infty )
\end{array}$$

Question 3

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

Accepted Answer

$\frac { 2 x h + x ^ { 2 } } { x }$

Question 4

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.

Accepted Answer

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

Question 5

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

Accepted Answer

The answer of Find the domain of \(f ( x...

Question 6

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

Accepted Answer

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

Question 7

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.

Accepted Answer

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)  $f ( x ) = \frac { 2 x ^ { 2 } + 1 } { x ^ { 2 } - 3 }$ 
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 8

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

Accepted Answer

The answer of Is \(g ( x ) = \frac...

Question 9

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

Accepted Answer

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: $[ 0 , \infty )$ 
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 10

Write the following expression as an algebraic expression that does not involve trigonometric or inverse trigonometric functions $\sin ^ { 2 } \left( \tan ^ { - 1 } 2 x \right)$

Accepted Answer

A)  $\frac { x ^ { 2 } } { 1 + 2 x ^ { 2 } }$ 
B)  $\frac { 2 x ^ { 2 } } { 1 + 2 x ^ { 2 } }$ 
C)  $\frac { 2 x ^ { 2 } } { 1 + 4 x ^ { 2 } }$ 
D)  $\frac { 4 x ^ { 2 } } { 1 + 4 x ^ { 2 } }$ 
A)  $\frac { x ^ { 2 } } { 1 + 2 x ^ { 2 } }$ 
B)  $\frac { 2 x ^ { 2 } } { 1 + 2 x ^ { 2 } }$ 
C)  $\frac { 2 x ^ { 2 } } { 1 + 4 x ^ { 2 } }$ 
D)  $\frac { 4 x ^ { 2 } } { 1 + 4 x ^ { 2 } }$

Question 11

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.

Accepted Answer

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

Question 12

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

Accepted Answer

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

Question 13

Which of the following is an odd function?

Accepted Answer

A)  $f ( x ) = x ^ { 3 } - x ^ { 2 }$ 
B)  $g ( x ) = \frac { 5 } { x }$ 
C)  $h ( x ) = x ^ { 4 } - x ^ { 2 }$ 
D)  $k ( x ) = | x - 2 |$ 
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 14

Solve the following equation for x $\tan ^ { 2 } \left( \sec ^ { - 1 } 2 x \right) = 0$

Accepted Answer

The answer of Solve the following equation for x \(\tan...

Question 15

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

Accepted Answer

\[\begin{array} { | l | l | l

Question 16

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

Accepted Answer

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: $\{ - 1,1 \}$ 
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 17

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

Accepted Answer

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

Question 18

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. $f ( g ( x ) )$

Accepted Answer

A. Even 
B

Question 19

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

Accepted Answer

The answer of Find the domain of \(f ( x...

Question 20

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.

Accepted Answer

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

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

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.

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

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.

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

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

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.

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

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

Write the following expression as an algebraic expression that does not involve trigonometric or inverse trigonometric functions $\sin ^ { 2 } \left( \tan ^ { - 1 } 2 x \right)$

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.

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

Which of the following is an odd function?

Solve the following equation for x $\tan ^ { 2 } \left( \sec ^ { - 1 } 2 x \right) = 0$

Table below defines two functions $f$ and $g$ Create an additional row for the table for the function $g ( x + 1 )$ x 0 1 2 3 4 5 6 f(x) 0 1 3 2 3 0 2 g(x) 1 0 1 1 0 1 0

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

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

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. $f ( g ( x ) )$

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

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.

Limits

Derivatives

Applications of the Derivative

Definite Integrals

Techniques of Integration

Applications of Integration

Sequences and Series

Power Series

Parametric Equations Polar Coordinates and Conic Sections

Vectors

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Multivariable Functions

Double and Triple Integrals

Vector Analysis

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Exam 15: Functions and Precalculus

Find the inverse of f(x)=1+4x3f ( x ) = \frac { 1 + 4 x } { 3 }f(x)=31+4x​

Given that f(x)=x2+2f ( x ) = x ^ { 2 } + 2f(x)=x2+2 , and g(x)=xg ( x ) = \sqrt { x }g(x)=x​ , find (f∘g)(x)( f \circ g ) ( x )(f∘g)(x) and (g∘f)(x)( g \circ f ) ( x )(g∘f)(x) , and their domains.

If g(x)=x2+1xg ( x ) = \frac { x ^ { 2 } + 1 } { x }g(x)=xx2+1​ , find g(x+h)−g(x)g ( x + h ) - g ( x )g(x+h)−g(x)

Find the domain of f(x)=1e3x−exf ( x ) = \frac { 1 } { e ^ { 3 x } - e ^ { x } }f(x)=e3x−ex1​

Let f(x)=x21+x3f ( x ) = \frac { x ^ { 2 } } { 1 + x ^ { 3 } }f(x)=1+x3x2​ Find XXX if f−1(x)=2f ^ { - 1 } ( x ) = 2f−1(x)=2

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

Is g(x)=x5−xx2+1g ( x ) = \frac { x ^ { 5 } - x } { x ^ { 2 } + 1 }g(x)=x2+1x5−x​ an even function, an odd function or neither? Explain.

Find the domain and range of f(x)=1x2+4f ( x ) = \frac { 1 } { x ^ { 2 } + 4 }f(x)=x2+41​

Write the following expression as an algebraic expression that does not involve trigonometric or inverse trigonometric functions sin⁡2(tan⁡−12x)\sin ^ { 2 } \left( \tan ^ { - 1 } 2 x \right)sin2(tan−12x)

A. If fff is an invertible function such that f(3)=3f ( 3 ) = 3f(3)=3 , then f−1(3)=13f ^ { - 1 } ( 3 ) = \frac { 1 } { 3 }f−1(3)=31​ . B. A one-to-one function is invertible. C. If fff and ggg are inverses of each other then fff and ggg have the same domain.

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

Which of the following is an odd function?

Solve the following equation for x tan⁡2(sec⁡−12x)=0\tan ^ { 2 } \left( \sec ^ { - 1 } 2 x \right) = 0tan2(sec−12x)=0

Table below defines two functions fff and ggg Create an additional row for the table for the function g(x+1)g ( x + 1 )g(x+1) x 0 1 2 3 4 5 6 f(x) 0 1 3 2 3 0 2 g(x) 1 0 1 1 0 1 0

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

Given y=1−xy = 1 - \sqrt { x }y=1−x​ , for what value(s) of XXX is y=−2y = - 2y=−2 ?

If fff is an odd function and ggg is an even function, determine whether the following functions are odd, even or neither. a. (f(x))2+(g(x))3( f ( x ) ) ^ { 2 } + ( g ( x ) ) ^ { 3 }(f(x))2+(g(x))3 b. 5+g(x)5 + g ( x )5+g(x) c. f(x3)f \left( x ^ { 3 } \right)f(x3) d. f(g(x))f ( g ( x ) )f(g(x))

Find the domain of f(x)=x2−4x2−9f ( x ) = \frac { \sqrt { x ^ { 2 } - 4 } } { x ^ { 2 } - 9 }f(x)=x2−9x2−4​​

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

Limits

Derivatives

Applications of the Derivative

Definite Integrals

Techniques of Integration

Applications of Integration

Sequences and Series

Power Series

Parametric Equations Polar Coordinates and Conic Sections

Vectors

Vector Functions

Multivariable Functions

Double and Triple Integrals

Vector Analysis

Filters

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

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.

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

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

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

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.

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

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

Write the following expression as an algebraic expression that does not involve trigonometric or inverse trigonometric functions $\sin ^ { 2 } \left( \tan ^ { - 1 } 2 x \right)$

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.

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

Solve the following equation for x $\tan ^ { 2 } \left( \sec ^ { - 1 } 2 x \right) = 0$

Table below defines two functions $f$ and $g$ Create an additional row for the table for the function $g ( x + 1 )$ x 0 1 2 3 4 5 6 f(x) 0 1 3 2 3 0 2 g(x) 1 0 1 1 0 1 0

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

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

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. $f ( g ( x ) )$

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

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.