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Why does the domain differ for different functions?
The domain of a function depends upon what values of the independent variable make the function undefined or imaginary.
How do we determine the domain of a function defined by an equation?
Explain why the domain of $\text{\hspace{0.17em}}f(x)=\sqrt[3]{x}\text{\hspace{0.17em}}$ is different from the domain of $\text{\hspace{0.17em}}f(x)=\sqrt[]{x}.$
There is no restriction on $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}f(x)=\sqrt[3]{x}\text{\hspace{0.17em}}$ because you can take the cube root of any real number. So the domain is all real numbers, $\text{\hspace{0.17em}}(-\infty ,\infty ).\text{\hspace{0.17em}}$ When dealing with the set of real numbers, you cannot take the square root of negative numbers. So $\text{\hspace{0.17em}}x$ -values are restricted for $\text{\hspace{0.17em}}f(x)=\sqrt[]{x}\text{\hspace{0.17em}}$ to nonnegative numbers and the domain is $\text{\hspace{0.17em}}[0,\infty ).$
When describing sets of numbers using interval notation, when do you use a parenthesis and when do you use a bracket?
How do you graph a piecewise function?
Graph each formula of the piecewise function over its corresponding domain. Use the same scale for the $\text{\hspace{0.17em}}x$ -axis and $\text{\hspace{0.17em}}y$ -axis for each graph. Indicate inclusive endpoints with a solid circle and exclusive endpoints with an open circle. Use an arrow to indicate $\text{\hspace{0.17em}}-\infty \text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\text{}\infty .\text{\hspace{0.17em}}$ Combine the graphs to find the graph of the piecewise function.
For the following exercises, find the domain of each function using interval notation.
$f(x)=-2x(x-1)(x-2)$
$f\left(x\right)=3\sqrt{x-2}$
$f(x)=\sqrt{4-3x}$
$\begin{array}{l}\\ f(x)=\sqrt[]{{x}^{2}+4}\end{array}$
$(-\infty ,\infty )$
$f(x)=\sqrt[3]{1-2x}$
$f(x)=\frac{9}{x-6}$
$f\left(x\right)=\frac{3x+1}{4x+2}$
$(-\infty ,-\frac{1}{2})\cup (-\frac{1}{2},\infty )$
$f\left(x\right)=\frac{\sqrt{x+4}}{x-4}$
$f(x)=\frac{x-3}{{x}^{2}+9x-22}$
$(-\infty ,-11)\cup (-11,2)\cup (2,\infty )$
$f(x)=\frac{1}{{x}^{2}-x-6}$
$f(x)=\frac{2{x}^{3}-250}{{x}^{2}-2x-15}$
$(-\infty ,-3)\cup (-3,5)\cup (5,\infty )$
$\frac{5}{\sqrt{x-3}}$
$f(x)=\frac{\sqrt{x-4}}{\sqrt{x-6}}$
$f(x)=\frac{x}{x}$
$f(x)=\frac{{x}^{2}-9x}{{x}^{2}-81}$
$\left(-\infty ,-9\right)\cup \left(-9,9\right)\cup \left(9,\infty \right)$
Find the domain of the function $\text{\hspace{0.17em}}f(x)=\sqrt{2{x}^{3}-50x}\text{\hspace{0.17em}}$ by:
For the following exercises, write the domain and range of each function using interval notation.
domain: $\text{\hspace{0.17em}}(2,8],\text{\hspace{0.17em}}$ range $\text{\hspace{0.17em}}[6,8)\text{\hspace{0.17em}}$
domain: $\text{\hspace{0.17em}}[-4,\text{4],}\text{\hspace{0.17em}}$ range: $\text{\hspace{0.17em}}[0,\text{2]}$
domain: $\text{\hspace{0.17em}}[-5,\text{}3),\text{\hspace{0.17em}}$ range: $\text{\hspace{0.17em}}\left[0,2\right]$
domain: $\text{\hspace{0.17em}}(-\infty ,1],\text{\hspace{0.17em}}$ range: $\text{\hspace{0.17em}}[0,\infty )\text{\hspace{0.17em}}$
domain: $\text{\hspace{0.17em}}\left[-6,-\frac{1}{6}\right]\cup \left[\frac{1}{6},6\right];\text{\hspace{0.17em}}$ range: $\text{\hspace{0.17em}}\left[-6,-\frac{1}{6}\right]\cup \left[\frac{1}{6},6\right]\text{\hspace{0.17em}}$
domain: $\text{\hspace{0.17em}}[-3,\text{}\infty );\text{\hspace{0.17em}}$ range: $\text{\hspace{0.17em}}[0,\infty )\text{\hspace{0.17em}}$
For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.
$f(x)=\{\begin{array}{lll}x+1\hfill & \text{if}\hfill & x<-2\hfill \\ -2x-3\hfill & \text{if}\hfill & x\ge -2\hfill \end{array}$
$f(x)=\{\begin{array}{lll}2x-1\hfill & \text{if}\hfill & x<1\hfill \\ 1+x\hfill & \text{if}\hfill & x\ge 1\hfill \end{array}$
domain: $\text{\hspace{0.17em}}(-\infty ,\infty )$
$f(x)=\{\begin{array}{c}x+1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x<0\\ x-1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x>0\end{array}$
$f\left(x\right)=\{\begin{array}{ccc}3& \text{if}& x<0\\ \sqrt{x}& \text{if}& x\ge 0\end{array}$
domain: $\text{\hspace{0.17em}}(-\infty ,\infty )$
$$f(x)=\{\begin{array}{c}{x}^{2}\text{if}x0\\ 1-x\text{if}x0\end{array}$$
$f(x)=\{\begin{array}{r}\hfill \begin{array}{r}\hfill {x}^{2}\\ \hfill x+2\end{array}\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{l}\text{if}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x<0\hfill \\ \text{if}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\ge 0\hfill \end{array}$
domain: $\text{\hspace{0.17em}}(-\infty ,\infty )$
$f\left(x\right)=\{\begin{array}{ccc}x+1& \text{if}& x<1\\ {x}^{3}& \text{if}& x\ge 1\end{array}$
$f(x)=\{\begin{array}{c}\left|x\right|\\ 1\end{array}\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x<2\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\ge 2\hfill \end{array}$
domain: $\text{\hspace{0.17em}}(-\infty ,\infty )$
For the following exercises, given each function $f,$ evaluate $f(\mathrm{-3}),\text{\hspace{0.17em}}f(\mathrm{-2}),\text{\hspace{0.17em}}f(\mathrm{-1}),$ and $f(0).$
$f(x)=\{\begin{array}{lll}x+1\hfill & \text{if}\hfill & x<-2\hfill \\ -2x-3\hfill & \text{if}\hfill & x\ge -2\hfill \end{array}$
$f(x)=\{\begin{array}{cc}1& \text{if}x\le -3\\ 0& \text{if}x-3\end{array}$
$\begin{array}{cccc}f(-3)=1;& f(-2)=0;& f(-1)=0;& f(0)=0\end{array}$
$f(x)=\{\begin{array}{cc}-2{x}^{2}+3& \text{if}x\le -1\\ 5x-7& \text{if}x-1\end{array}$
For the following exercises, given each function $\text{\hspace{0.17em}}f,\text{\hspace{0.17em}}$ evaluate $f(\mathrm{-1}),\text{\hspace{0.17em}}f(0),\text{\hspace{0.17em}}f(2),\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f(4).$
$f(x)=\{\begin{array}{lll}7x+3\hfill & \text{if}\hfill & x<0\hfill \\ 7x+6\hfill & \text{if}\hfill & x\ge 0\hfill \end{array}$
$\begin{array}{cccc}f(-1)=-4;& f(0)=6;& f(2)=20;& f(4)=34\end{array}$
$f\left(x\right)=\{\begin{array}{ccc}{x}^{2}-2& \text{if}& x<2\\ 4+\left|x-5\right|& \text{if}& x\ge 2\end{array}$
$f\left(x\right)=\{\begin{array}{ccc}5x& \text{if}& x<0\\ 3& \text{if}& 0\le x\le 3\\ {x}^{2}& \text{if}& x>3\end{array}$
$\begin{array}{cccc}f(-1)=-5;& f(0)=3;& f(2)=3;& f(4)=16\end{array}$
For the following exercises, write the domain for the piecewise function in interval notation.
$f(x)=\{\begin{array}{c}x+1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x-2\\ -2x-3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\ge -2\end{array}$
$f(x)=\{\begin{array}{c}{x}^{2}-2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x1\\ -{x}^{2}+2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x1\end{array}$
domain: $\text{\hspace{0.17em}}(-\infty ,1)\cup (1,\infty )$
$f(x)=\{\begin{array}{c}2x-3\\ -3{x}^{2}\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}\text{if}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x<0\\ \text{if}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\ge 2\end{array}$
Graph $\text{\hspace{0.17em}}y=\frac{1}{{x}^{2}}\text{\hspace{0.17em}}$ on the viewing window $\text{\hspace{0.17em}}[\mathrm{-0.5},\mathrm{-0.1}]\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}[0.1,0.5].\text{\hspace{0.17em}}$ Determine the corresponding range for the viewing window. Show the graphs.
window: $\text{\hspace{0.17em}}[-0.5,-0.1];\text{\hspace{0.17em}}$ range: $\text{\hspace{0.17em}}[4,\text{}100]$
window: $\text{\hspace{0.17em}}[0.1,\text{}0.5];\text{\hspace{0.17em}}$ range: $\text{\hspace{0.17em}}[4,\text{}100]$
Graph $\text{\hspace{0.17em}}y=\frac{1}{x}\text{\hspace{0.17em}}$ on the viewing window $\text{\hspace{0.17em}}[\mathrm{-0.5},\mathrm{-0.1}]\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}[0.1,\text{}0.5].\text{\hspace{0.17em}}$ Determine the corresponding range for the viewing window. Show the graphs.
Suppose the range of a function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}[\mathrm{-5},\text{}8].\text{\hspace{0.17em}}$ What is the range of $\text{\hspace{0.17em}}|f(x)|?$
$[0,\text{}8]$
Create a function in which the range is all nonnegative real numbers.
Create a function in which the domain is $\text{\hspace{0.17em}}x>2.$
Many answers. One function is $\text{\hspace{0.17em}}f(x)=\frac{1}{\sqrt{x-2}}.$
The height $\text{\hspace{0.17em}}h\text{\hspace{0.17em}}$ of a projectile is a function of the time $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ it is in the air. The height in feet for $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ seconds is given by the function $h(t)=\mathrm{-16}{t}^{2}+96t.$ What is the domain of the function? What does the domain mean in the context of the problem?
The domain is $\text{\hspace{0.17em}}[0,\text{}6];\text{\hspace{0.17em}}$ it takes 6 seconds for the projectile to leave the ground and return to the ground
The cost in dollars of making $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ items is given by the function $\text{\hspace{0.17em}}C(x)=10x+500.$
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