Exam 3: Polynomial and Rational Functions

Sketch the graph of the function. $f ( x ) = \left\{ \begin{array} { l l } x ^ { 2 } & \text { if } x < - 3 \\x + 12 & \text { if } x \geq - 3\end{array} \right.$

(Essay)

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Question 41

Determine whether $f ( x ) = x ^ { 2 } - x ^ { 4 }$ is even, odd, or neither

(Multiple Choice)

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Question 42

If an object is dropped from a high cliff or a tall building, then the distance it has fallen after t seconds is given by the function $f ( t ) = 16 t ^ { 2 }$ Find its average speed (average rate of change) over the following intervals: (i) Between 2 s and 7 s (ii) Between $t = c \text { and } t = c + h$

(Essay)

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Question 43

Find $g \circ g \circ g$ , where $g ( x ) = x ^ { 2 }$ .

(Multiple Choice)

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Question 44

A function is given. Use a graphing calculator to draw the graph of f. Find the domain and range of f from the graph. $f ( x ) = - \sqrt { 16 - x ^ { 2 } }$

(Short Answer)

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Question 45

If $g ( x ) = \frac { 4 } { x } + x ^ { 2 }$ , find $g ( 4 )$

(Multiple Choice)

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Question 46

Determine whether the function $f ( x ) = - 2 x ^ { 2 } + 18 x - 16$ is one-to-one.

(Essay)

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Question 47

Use a graphing calculator or computer to determine whether the function $f ( x ) = - | x | - | 5 - x |$ is one-to-one.

(Essay)

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Question 48

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 each answer correct to two decimal places. $U ( x ) = 4 \left( x ^ { 3 } - x \right)$

(Essay)

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Question 49

Use a graphing calculator to find, approximately the range of the function. $f ( x ) = 2 x ^ { 4 } - x ^ { 3 } + x ^ { 2 } + 2 x - 7$

(Multiple Choice)

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Question 50

Given $f ( x ) = 2 + x ^ { 2 }$ and $g ( x ) = \sqrt { x - 1 }$ , find $( f \circ g ) ( 2 )$ , $( f \circ f ) ( 2 )$

(Short Answer)

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Question 51

Find the domain of the function. $f ( x ) = 3 x - \frac { 2 } { \sqrt { x + 1 } }$

(Multiple Choice)

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Question 52

Find the domain of the function. $f ( x ) = 2 x ^ { 2 } - 3 , \quad 0 \leq x < 5$

(Multiple Choice)

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Question 53

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. Estimate the average rate of change of W between x = 0 and x= 100?

(Multiple Choice)

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Question 54

If an object is dropped from a high cliff or a tall building, then the distance it has fallen after t seconds is given by the function $f ( t ) = 16 t ^ { 2 }$ Find its average speed (average rate of change) over the interval between $t = c \text { and } t = c + h$

(Multiple Choice)

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Question 55

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

(Essay)

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Question 56

Sketch the graph of the function. $h ( x ) = x ^ { 3 } - 4 x ^ { 2 }$

(Multiple Choice)

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Question 57

Sketch the graph of the function. $f(x)=\left\{\begin{array}{ll}-2 x & \text { if } x<0 \\1-2 x & \text { if } x \geq 0\end{array}\right.$

(Multiple Choice)

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Question 58

Describe how the graph of $y = - f ( 3 x ) + 4$ can be obtained from the graph of $f .$

(Essay)

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Question 59

Sketch the graph of the function $f ( x ) = 2 + \sqrt { x }$ .

(Essay)

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Question 60

Sketch the graph of the function. $f ( x ) = \left\{ \begin{array} { l l } x ^ { 2 } & \text { if } x < - 3 \\x + 12 & \text { if } x \geq - 3\end{array} \right.$

Determine whether $f ( x ) = x ^ { 2 } - x ^ { 4 }$ is even, odd, or neither

Find $g \circ g \circ g$ , where $g ( x ) = x ^ { 2 }$ .

A function is given. Use a graphing calculator to draw the graph of f. Find the domain and range of f from the graph. $f ( x ) = - \sqrt { 16 - x ^ { 2 } }$

If $g ( x ) = \frac { 4 } { x } + x ^ { 2 }$ , find $g ( 4 )$

Determine whether the function $f ( x ) = - 2 x ^ { 2 } + 18 x - 16$ is one-to-one.

Use a graphing calculator or computer to determine whether the function $f ( x ) = - | x | - | 5 - x |$ is one-to-one.

Use a graphing calculator to find, approximately the range of the function. $f ( x ) = 2 x ^ { 4 } - x ^ { 3 } + x ^ { 2 } + 2 x - 7$

Given $f ( x ) = 2 + x ^ { 2 }$ and $g ( x ) = \sqrt { x - 1 }$ , find $( f \circ g ) ( 2 )$ , $( f \circ f ) ( 2 )$

Find the domain of the function. $f ( x ) = 3 x - \frac { 2 } { \sqrt { x + 1 } }$

Find the domain of the function. $f ( x ) = 2 x ^ { 2 } - 3 , \quad 0 \leq x < 5$

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. Estimate the average rate of change of W between x = 0 and x= 100?

If an object is dropped from a high cliff or a tall building, then the distance it has fallen after t seconds is given by the function $f ( t ) = 16 t ^ { 2 }$ Find its average speed (average rate of change) over the interval between $t = c \text { and } t = c + h$

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

Sketch the graph of the function. $h ( x ) = x ^ { 3 } - 4 x ^ { 2 }$

Sketch the graph of the function. $f(x)=\left\{\begin{array}{ll}-2 x & \text { if } x<0 \\1-2 x & \text { if } x \geq 0\end{array}\right.$

Describe how the graph of $y = - f ( 3 x ) + 4$ can be obtained from the graph of $f .$

Sketch the graph of the function $f ( x ) = 2 + \sqrt { x }$ .

Equations and Graphs

Functions

Exponential and Logarithmic Functions

Trigonometric Functions: Right Triangle Approach

Trigonometric Functions: Unit Circle Approach

Conic Sections

Polar Coordinates and Parametric Equations

Vectors in Two and Three Dimensions

Systems of Equations and Inequalities

Matrices and Determinants

Conic Sections

Sequences and Series

Counting and Probability

Filters

Exam 3: Polynomial and Rational Functions

Sketch the graph of the function. f(x)={x2 if x<−3x+12 if x≥−3f ( x ) = \left\{ \begin{array} { l l } x ^ { 2 } & \text { if } x < - 3 \\x + 12 & \text { if } x \geq - 3\end{array} \right.f(x)={x2x+12​ if x<−3 if x≥−3​

Determine whether f(x)=x2−x4f ( x ) = x ^ { 2 } - x ^ { 4 }f(x)=x2−x4 is even, odd, or neither

Find g∘g∘gg \circ g \circ gg∘g∘g , where g(x)=x2g ( x ) = x ^ { 2 }g(x)=x2 .

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)=−16−x2f ( x ) = - \sqrt { 16 - x ^ { 2 } }f(x)=−16−x2​

If g(x)=4x+x2g ( x ) = \frac { 4 } { x } + x ^ { 2 }g(x)=x4​+x2 , find g(4)g ( 4 )g(4)

Determine whether the function f(x)=−2x2+18x−16f ( x ) = - 2 x ^ { 2 } + 18 x - 16f(x)=−2x2+18x−16 is one-to-one.

Use a graphing calculator or computer to determine whether the function f(x)=−∣x∣−∣5−x∣f ( x ) = - | x | - | 5 - x |f(x)=−∣x∣−∣5−x∣ is one-to-one.

Use a graphing calculator to find, approximately the range of the function. f(x)=2x4−x3+x2+2x−7f ( x ) = 2 x ^ { 4 } - x ^ { 3 } + x ^ { 2 } + 2 x - 7f(x)=2x4−x3+x2+2x−7

Given f(x)=2+x2f ( x ) = 2 + x ^ { 2 }f(x)=2+x2 and g(x)=x−1g ( x ) = \sqrt { x - 1 }g(x)=x−1​ , find (f∘g)(2)( f \circ g ) ( 2 )(f∘g)(2) , (f∘f)(2)( f \circ f ) ( 2 )(f∘f)(2)

Find the domain of the function. f(x)=3x−2x+1f ( x ) = 3 x - \frac { 2 } { \sqrt { x + 1 } }f(x)=3x−x+1​2​

Find the domain of the function. f(x)=2x2−3,0≤x<5f ( x ) = 2 x ^ { 2 } - 3 , \quad 0 \leq x < 5f(x)=2x2−3,0≤x<5

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. Estimate the average rate of change of W between x = 0 and x= 100?

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

Sketch the graph of the function. h(x)=x3−4x2h ( x ) = x ^ { 3 } - 4 x ^ { 2 }h(x)=x3−4x2

Sketch the graph of the function. f(x)={−2x if x<01−2x if x≥0f(x)=\left\{\begin{array}{ll}-2 x & \text { if } x<0 \\1-2 x & \text { if } x \geq 0\end{array}\right.f(x)={−2x1−2x​ if x<0 if x≥0​

Describe how the graph of y=−f(3x)+4y = - f ( 3 x ) + 4y=−f(3x)+4 can be obtained from the graph of f.f .f.

Sketch the graph of the function f(x)=2+xf ( x ) = 2 + \sqrt { x }f(x)=2+x​ .

Equations and Graphs

Functions

Exponential and Logarithmic Functions

Trigonometric Functions: Right Triangle Approach

Trigonometric Functions: Unit Circle Approach

Conic Sections

Polar Coordinates and Parametric Equations

Vectors in Two and Three Dimensions

Systems of Equations and Inequalities

Matrices and Determinants

Conic Sections

Sequences and Series

Counting and Probability

Filters

Sketch the graph of the function. $f ( x ) = \left\{ \begin{array} { l l } x ^ { 2 } & \text { if } x < - 3 \\x + 12 & \text { if } x \geq - 3\end{array} \right.$

Determine whether $f ( x ) = x ^ { 2 } - x ^ { 4 }$ is even, odd, or neither

Find $g \circ g \circ g$ , where $g ( x ) = x ^ { 2 }$ .

A function is given. Use a graphing calculator to draw the graph of f. Find the domain and range of f from the graph. $f ( x ) = - \sqrt { 16 - x ^ { 2 } }$

If $g ( x ) = \frac { 4 } { x } + x ^ { 2 }$ , find $g ( 4 )$

Determine whether the function $f ( x ) = - 2 x ^ { 2 } + 18 x - 16$ is one-to-one.

Use a graphing calculator or computer to determine whether the function $f ( x ) = - | x | - | 5 - x |$ is one-to-one.

Use a graphing calculator to find, approximately the range of the function. $f ( x ) = 2 x ^ { 4 } - x ^ { 3 } + x ^ { 2 } + 2 x - 7$

Given $f ( x ) = 2 + x ^ { 2 }$ and $g ( x ) = \sqrt { x - 1 }$ , find $( f \circ g ) ( 2 )$ , $( f \circ f ) ( 2 )$

Find the domain of the function. $f ( x ) = 3 x - \frac { 2 } { \sqrt { x + 1 } }$

Find the domain of the function. $f ( x ) = 2 x ^ { 2 } - 3 , \quad 0 \leq x < 5$

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

Sketch the graph of the function. $h ( x ) = x ^ { 3 } - 4 x ^ { 2 }$

Sketch the graph of the function. $f(x)=\left\{\begin{array}{ll}-2 x & \text { if } x<0 \\1-2 x & \text { if } x \geq 0\end{array}\right.$

Describe how the graph of $y = - f ( 3 x ) + 4$ can be obtained from the graph of $f .$

Sketch the graph of the function $f ( x ) = 2 + \sqrt { x }$ .