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Why does the horizontal line test tell us whether the graph of a function is one-to-one?
When a horizontal line intersects the graph of a function more than once, that indicates that for that output there is more than one input. A function is one-to-one if each output corresponds to only one input.
For the following exercises, determine whether the relation represents a function.
$\left\{\left(a,b\right),\text{}\left(c,d\right),\text{}\left(a,c\right)\right\}$
For the following exercises, determine whether the relation represents $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ as a function of $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$
$5x+2y=10$
$x={y}^{2}$
$2x+{y}^{2}=6$
$y=\frac{1}{x}$
$x=\sqrt{1-{y}^{2}}$
${x}^{2}+{y}^{2}=9$
$x={y}^{3}$
$y=\sqrt{1-{x}^{2}}$
$y=\pm \sqrt{1-x}$
${y}^{3}={x}^{2}$
For the following exercises, evaluate the function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ at the indicated values $\text{}f(\mathrm{-3}),f(2),f(-a),-f(a),f(a+h).$
$f(x)=2x-5$
$\begin{array}{cccc}f(-3)=-11;& f(2)=-1;& f(-a)=-2a-5;& -f(a)=-2a+5;\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}f(a+h)=2a+2h-5\end{array}$
$f(x)=-5{x}^{2}+2x-1$
$f(x)=\sqrt{2-x}+5$
$\begin{array}{cccc}f(-3)=\sqrt{5}+5;& f(2)=5;& f(-a)=\sqrt{2+a}+5;& -f(a)=-\sqrt{2-a}-5;\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}f(a+h)=\end{array}$ $\sqrt{2-a-h}+5$
$f(x)=\frac{6x-1}{5x+2}$
$f(x)=\left|x-1\right|-\left|x+1\right|$
$\begin{array}{cccc}f(-3)=2;& f(2)=1-3=-2;& f(-a)=\left|-a-1\right|-\left|-a+1\right|;& -f(a)=-\left|a-1\right|\text{\hspace{0.17em}}+\left|a+1\right|;\text{\hspace{1em}}\text{}f(a+h)=\text{\hspace{0.17em}}\left|a+h-1\right|-\left|a+h+1\right|\end{array}$
Given the function $\text{\hspace{0.17em}}g(x)=5-{x}^{2},\text{\hspace{0.17em}}$ simplify $\text{\hspace{0.17em}}\frac{g(x+h)-g(x)}{h},\text{\hspace{0.17em}}h\ne 0.$
Given the function $\text{\hspace{0.17em}}g(x)={x}^{2}+2x,\text{\hspace{0.17em}}$ simplify $\text{\hspace{0.17em}}\frac{g(x)-g(a)}{x-a},\text{\hspace{0.17em}}x\ne a.$
$\frac{g(x)-g(a)}{x-a}=x+a+2,\text{\hspace{0.17em}}x\ne a$
Given the function $\text{\hspace{0.17em}}k(t)=2t-1\text{:}$
Given the function $\text{\hspace{0.17em}}f(x)=8-3x\text{:}$
a. $\text{\hspace{0.17em}}f(-2)=14;\text{\hspace{0.17em}}$ b. $\text{\hspace{0.17em}}x=3$
Given the function $\text{\hspace{0.17em}}p(c)={c}^{2}+c\text{:}$
Given the function $\text{\hspace{0.17em}}f(x)={x}^{2}-3x\text{:}$
a. $\text{\hspace{0.17em}}f(5)=10;\text{\hspace{0.17em}}$ b. $\text{\hspace{0.17em}}x=-1\text{}$ or $\text{}x=4$
Given the function $\text{\hspace{0.17em}}f(x)=\sqrt{x+2}\text{:}$
Consider the relationship $\text{\hspace{0.17em}}3r+2t=18.$
a. $\text{\hspace{0.17em}}f(t)=6-\frac{2}{3}t;\text{\hspace{0.17em}}$ b. $\text{\hspace{0.17em}}f(-3)=8;\text{\hspace{0.17em}}$ c. $\text{\hspace{0.17em}}t=6\text{\hspace{0.17em}}$
For the following exercises, use the vertical line test to determine which graphs show relations that are functions.
Given the following graph,
Given the following graph,
a. $\text{\hspace{0.17em}}f(0)=1;\text{\hspace{0.17em}}$ b. $\text{\hspace{0.17em}}f(x)=-3,\text{\hspace{0.17em}}x=-2\text{}$ or $\text{}x=2\text{\hspace{0.17em}}$
Given the following graph,
For the following exercises, determine if the given graph is a one-to-one function.
For the following exercises, determine whether the relation represents a function.
$\left\{\left(\mathrm{-1},\mathrm{-1}\right),\left(\mathrm{-2},\mathrm{-2}\right),\left(\mathrm{-3},\mathrm{-3}\right)\right\}$
$\left\{\left(3,4\right),\left(4,5\right),\left(5,6\right)\right\}$
function
$\left\{(2,5),(7,11),(15,8),(7,9)\right\}$
For the following exercises, determine if the relation represented in table form represents $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ as a function of $\text{\hspace{0.17em}}x.$
$x$ | 5 | 10 | 15 |
$y$ | 3 | 8 | 8 |
For the following exercises, use the function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ represented in [link] .
$x$ | $f\left(x\right)$ |
0 | 74 |
1 | 28 |
2 | 1 |
3 | 53 |
4 | 56 |
5 | 3 |
6 | 36 |
7 | 45 |
8 | 14 |
9 | 47 |
Evaluate $\text{\hspace{0.17em}}f(3).$
Solve $\text{\hspace{0.17em}}f(x)=1.$
$f(x)=1,\text{\hspace{0.17em}}x=2$
For the following exercises, evaluate the function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ at the values $f\left(-2\right),\text{\hspace{0.17em}}f(\mathrm{-1}),\text{\hspace{0.17em}}f(0),\text{\hspace{0.17em}}f(1),$ and $\text{\hspace{0.17em}}f(2).$
$f\left(x\right)=4-2x$
$f\left(x\right)=8-3x$
$\begin{array}{ccccc}f(-2)=14;& f(-1)=11;& f(0)=8;& f(1)=5;& f(2)=2\end{array}$
$f\left(x\right)=8{x}^{2}-7x+3$
$f\left(x\right)=3+\sqrt{x+3}$
$\begin{array}{ccccc}f(-2)=4;\text{\hspace{1em}\hspace{1em}}& f(-1)=4.414;& f(0)=4.732;& f(1)=4.5;& f(2)=5.236\end{array}$
$f(x)=\frac{x-2}{x+3}$
$f\left(x\right)={3}^{x}$
$\begin{array}{ccccc}f(-2)=\frac{1}{9};& f(-1)=\frac{1}{3};& f(0)=1;& f(1)=3;& f(2)=9\end{array}$
For the following exercises, evaluate the expressions, given functions $f,\text{\hspace{0.17em}}\text{\hspace{0.17em}}g,$ and $\text{\hspace{0.17em}}h\text{:}$
$3f\left(1\right)-4g\left(-2\right)$
For the following exercises, graph $\text{\hspace{0.17em}}y={x}^{2}\text{\hspace{0.17em}}$ on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.
$[-0.1,\text{}0.1]$
$[-100,100]$
For the following exercises, graph $\text{\hspace{0.17em}}y={x}^{3}\text{\hspace{0.17em}}$ on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.
$[-10,\text{10}]$
For the following exercises, graph $\text{\hspace{0.17em}}y=\sqrt{x}\text{\hspace{0.17em}}$ on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.
$[0,\text{0}\text{.01}]$
$[0,\text{10,000}]$
For the following exercises, graph $y=\sqrt[3]{x}$ on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.
$[\mathrm{-0.001},\text{0.001}]$
$[\mathrm{-0.1},\text{0.1}]$
$[\mathrm{-1000},\text{1000}]$
$[\mathrm{-1,000,000},\text{1,000,000}]$
$[-100,\text{100}]$
The amount of garbage, $\text{\hspace{0.17em}}G,\text{\hspace{0.17em}}$ produced by a city with population $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ is given by $\text{\hspace{0.17em}}G=f\left(p\right).\text{\hspace{0.17em}}$ $G\text{\hspace{0.17em}}$ is measured in tons per week, and $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ is measured in thousands of people.
The number of cubic yards of dirt, $\text{\hspace{0.17em}}D,\text{\hspace{0.17em}}$ needed to cover a garden with area $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ square feet is given by $\text{\hspace{0.17em}}D=g\left(a\right).$
a. $\text{\hspace{0.17em}}g(5000)=50;$ b. The number of cubic yards of dirt required for a garden of 100 square feet is 1.
Let $\text{\hspace{0.17em}}f\left(t\right)\text{\hspace{0.17em}}$ be the number of ducks in a lake $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ years after 1990. Explain the meaning of each statement:
Let $\text{\hspace{0.17em}}h\left(t\right)\text{\hspace{0.17em}}$ be the height above ground, in feet, of a rocket $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ seconds after launching. Explain the meaning of each statement:
a. The height of a rocket above ground after 1 second is 200 ft. b. the height of a rocket above ground after 2 seconds is 350 ft.
Show that the function $\text{\hspace{0.17em}}f\left(x\right)=3{\left(x-5\right)}^{2}+7\text{\hspace{0.17em}}$ is not one-to-one.
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