Find the domain and range of each function.

$$f(x)=1-\sqrt{x}$$

Andrew B.

Numerade Educator

Find the domain and range of each function.

$$F(x)=\sqrt{5 x+10}$$

Evan L.

Numerade Educator

Find the domain and range of each function.

$$g(x)=\sqrt{x^{2}-3 x}$$

Andrew B.

Numerade Educator

Find the domain and range of each function.

$$f(t)=\frac{4}{3-t}$$

Evan L.

Numerade Educator

Find the domain and range of each function.

$$G(t)=\frac{2}{t^{2}-16}$$

Andrew B.

Numerade Educator

Which of the graphs are graphs of functions of $x$ and which are not? Give reasons for your answers.

(GRAPH CAN'T COPY)

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(GRAPH CAN'T COPY)

Andrew B.

Numerade Educator

Express the area and perimeter of an equilateral triangle as a function of the triangle's side length $x$.

Evan L.

Numerade Educator

Express the side length of a square as a function of the length $d$ of the square's diagonal. Then express the area as a function of the diagonal length.

Andrew B.

Numerade Educator

Express the edge length of a cube as a function of the cube's diagonal length $d$. Then express the surface area and volume of the cube as a function of the diagonal length.

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A point $P$ in the first quadrant lies on the graph of the function $f(x)=\sqrt{x} .$ Express the coordinates of $P$ as functions of the slope of the line joining $P$ to the origin.

Andrew B.

Numerade Educator

Consider the point $(x, y)$ lying on the graph of the line $2 x+4 y=5$ Let $L$ be the distance from the point $(x, y)$ to the origin $(0,0) .$ Write $L$ as a function of $x$

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Consider the point $(x, y)$ lying on the graph of $y=\sqrt{x-3}$. Let L be the distance between the points $(x, y)$ and $(4,0) .$ Write $L$ as a function of $y$

Andrew B.

Numerade Educator

Find the natural domain and graph the functions.

$$f(x)=1-2 x-x^{2}$$

Andrew B.

Numerade Educator

Find the natural domain and graph the functions.

$$g(x)=\sqrt{|x|}$$

Evan L.

Numerade Educator

Find the natural domain and graph the functions.

$$g(x)=\sqrt{-x}$$

Andrew B.

Numerade Educator

Find the natural domain and graph the functions.

$$F(t)=t /|t|$$

Evan L.

Numerade Educator

Find the natural domain and graph the functions.

$$G(t)=1 /|t|$$

Andrew B.

Numerade Educator

Graph the following equations and explain why they are not graphs of functions of $x$

a. $|y|=x$

b. $y^{2}=x^{2}$

Evan L.

Numerade Educator

Graph the following equations and explain why they are not graphs of functions of $x$

a. $|x|+|y|=1$

b. $|x+y|=1$

Andrew B.

Numerade Educator

Graph the functions.

$$f(x)=\left\{\begin{array}{ll}

x, & 0 \leq x \leq 1 \\

2-x, & 1<x \leq 2

\end{array}\right.$$

Evan L.

Numerade Educator

Graph the functions.

$$g(x)=\left\{\begin{array}{ll}

1-x, & 0 \leq x \leq 1 \\

2-x, & 1<x \leq 2

\end{array}\right.$$

Andrew B.

Numerade Educator

Graph the functions.

$$F(x)=\left\{\begin{array}{ll}

4-x^{2}, & x \leq 1 \\

x^{2}+2 x, & x>1

\end{array}\right.$$

Evan L.

Numerade Educator

Graph the functions.

$$G(x)=\left\{\begin{array}{ll}

1 / x, & x<0 \\

x, & 0 \leq x

\end{array}\right.$$

Andrew B.

Numerade Educator

What real numbers $x$ satisfy the equation $\lfloor x\rfloor=\lceil x | ?$

Andrew B.

Numerade Educator

Does $\lceil-x\rceil=-\lfloor x\rfloor$ for all real $x ?$ Give reasons for your answer.

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Graph the function

$$

f(x)=\left\{\begin{array}{ll}

\lfloor x\rfloor, & x \geq 0 \\

| x\rceil, & x<0

\end{array}\right.

$$

Why is $f(x)$ called the integer part of $x ?$

Andrew B.

Numerade Educator

Graph the functions. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.

$$y=-x^{3}$$

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Graph the functions. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.

$$y=-\frac{1}{x^{2}}$$

Andrew B.

Numerade Educator

Graph the functions. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.

$$y=-\frac{1}{x}$$

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Graph the functions. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.

$$y=\frac{1}{|x|}$$

Andrew B.

Numerade Educator

Graph the functions. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.

$$y=\sqrt{|x|}$$

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Graph the functions. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.

$$y=\sqrt{-x}$$

Andrew B.

Numerade Educator

Graph the functions. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.

$$y=x^{3} / 8$$

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Graph the functions. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.

$$y=-4 \sqrt{x}$$

Andrew B.

Numerade Educator

Graph the functions. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.

$$y=-x^{3 / 2}$$

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Graph the functions. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.

$$y=(-x)^{2 / 3}$$

Andrew B.

Numerade Educator

Say whether the function is even, odd, or neither. Give reasons for your answer.

$$f(x)=3$$

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Say whether the function is even, odd, or neither. Give reasons for your answer.

$$f(x)=x^{-5}$$

Andrew B.

Numerade Educator

Say whether the function is even, odd, or neither. Give reasons for your answer.

$$f(x)=x^{2}+1$$

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Say whether the function is even, odd, or neither. Give reasons for your answer.

$$f(x)=x^{2}+x$$

Andrew B.

Numerade Educator

Say whether the function is even, odd, or neither. Give reasons for your answer.

$$g(x)=x^{3}+x$$

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Say whether the function is even, odd, or neither. Give reasons for your answer.

$$g(x)=x^{4}+3 x^{2}-1$$

Andrew B.

Numerade Educator

Say whether the function is even, odd, or neither. Give reasons for your answer.

$$g(x)=\frac{1}{x^{2}-1}$$

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Say whether the function is even, odd, or neither. Give reasons for your answer.

$$g(x)=\frac{x}{x^{2}-1}$$

Andrew B.

Numerade Educator

Say whether the function is even, odd, or neither. Give reasons for your answer.

$$h(t)=\frac{1}{t-1}$$

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Say whether the function is even, odd, or neither. Give reasons for your answer.

$$h(t)=\left|t^{3}\right|$$

Andrew B.

Numerade Educator

Say whether the function is even, odd, or neither. Give reasons for your answer.

$$h(t)=2 t+1$$

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Say whether the function is even, odd, or neither. Give reasons for your answer.

$$h(t)=2|t|+1$$

Andrew B.

Numerade Educator

Say whether the function is even, odd, or neither. Give reasons for your answer.

$$\sin 2 x$$

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Say whether the function is even, odd, or neither. Give reasons for your answer.

$$\sin x^{2}$$

Andrew B.

Numerade Educator

Say whether the function is even, odd, or neither. Give reasons for your answer.

$$\cos 3 x$$

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Say whether the function is even, odd, or neither. Give reasons for your answer.

$$1+\cos x$$

Andrew B.

Numerade Educator

The variable $s$ is proportional to $t$, and $s=25$ when $t=75 .$ Determine $t$ when $s=69$

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The kinetic energy $K$ of a mass is proportional to the square of its velocity $v .$ If $K=12,960$ joules when $v=18 \mathrm{m} / \mathrm{scc},$ what is $K$ when $v=10 \mathrm{m} / \mathrm{sec} ?$

Andrew B.

Numerade Educator

The variables $r$ and $s$ are inversely proportional, and $r=6$ when $s=4 .$ Determine $s$ when $r=10$

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Boyle's Law says that the volume $V$ of a gas at constant temperature increases whenever the pressure $P$ decreases, so that $V$ and $P$ are inversely proportional. If $P=14.7 \mathrm{lb} / \mathrm{in}^{2}$ when $V=1000 \mathrm{in}^{3},$ then what is $V$ when $P=23.4 \mathrm{lb} / \mathrm{in}^{2} ?$

Andrew B.

Numerade Educator

A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 14 in. by 22 in. by cutting out equal squares of side $x$ at each corner and then folding up the sides as in the figure. Express the volume $V$ of the box as a function of $x$.

(FIGURE CAN'T COPY)

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The accompanying figure shows a rectangle inscribed in an isosceles right triangle whose hypotenuse is 2 units long.

a. Express the $y$ -coordinate of $P$ in terms of $x$. (You might start by writing an equation for the line $A B .$ )

b. Express the area of the rectangle in terms of $x$.

Andrew B.

Numerade Educator

Match each equation with its graph. Do not use a graphing device, and give reasons for your answer.

(GRAPH CAN'T COPY)

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(GRAPH CAN'T COPY)

Andrew B.

Numerade Educator

a. Graph the functions $f(x)=x / 2$ and $g(x)=1+(4 / x)$ together to identify the values of $x$ for which

$$

\frac{\sqrt{3}}{2}+1 \frac{4}{4}

$$

b. Confirm your findings in part (a) algebraically.

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a. Graph the functions $f(x)=3 /(x-1)$ and $g(x)=2 /(x+1)$ together to identify the values of $x$ for which

$$

\frac{3}{x-1}<\frac{2}{x+1}

$$

b. Confirm your findings in part (a) algebraically.

Andrew B.

Numerade Educator

For a curve to be symmetric about the $x$ -axis, the point $(x, y)$ must lie on the curve if and only if the point $(x,-y)$ lies on the curve. Explain why a curve that is symmetric about the $x$ -axis is not the graph of a function, unless the function is $y=0$

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Three hundred books sell for $\$ 40$ each, resulting in a revenue of $(300)(\$ 40)=\$ 12,000 .$ For each $\$ 5$ increase in the price, 25 fewer books are sold. Write the revenue $R$ as a function of the number $x$ of $\$ 5$ increases.

Andrew B.

Numerade Educator

A pen in the shape of an isosceles right triangle with legs of length $x$ ft and hypotenuse of length $h$ ft is to be built. If fencing costs S5/ft for the legs and $\$ 10 /$ ft for the hypotenuse, write the total cost $C$ of construction as a function of $h$

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Industrial costs A power plant sits next to a river where the river is $800 \mathrm{ft}$ wide. To lay a new cable from the plant to a location in the city 2 mi downstream on the opposite side costs $\$ 180$ per foot across the river and $\$ 100$ per foot along the land.a. Suppose that the cable goes from the plant to a point $Q$ on the opposite side that is $x$ ft from the point $P$ directly opposite the plant. Write a function $C(x)$ that gives the cost of laying the cable in terms of the distance $x$.

b. Generate a table of values to determine whether the least expensive location for point $Q$ is less than 2000 ft or greater than $2000 \mathrm{ft}$ from point $P$

Andrew B.

Numerade Educator