함수의 뜻

For the following exercises, given each function $f,$ evaluate $f(-3),\,f(-2),\,f(-1),$ and $f(0).$

$f(x)=\{\begin{array}{cc}-2{x}^{2}+3 & \text{if }x\le -1 \\ 5x-7 & \text{if }x>-1\end{array}$

For the following exercises, given each function $f,$ evaluate $f(-1),\,f(0),\,f(2),$ and $f(4).$

$f(x)=\{\begin{array}{ccc}7x+3 & \text{if} & x<0 \\ 7x+6 & \text{if} & x\ge 0\end{array}$

For the following exercises, given each function $f,$ evaluate $f(-1),\,f(0),\,f(2),$ and $f(4).$

$f(x)=\{\begin{array}{ccc}{x}^{2}-2 & \text{if} & x<2 \\ 4+|x-5| & \text{if} & x\ge 2\end{array}$

For the following exercises, given each function $f,$ evaluate $f(-1),\,f(0),\,f(2),$ and $f(4).$

$f(x)=\{\begin{array}{ccc}5x & \text{if} & x<0 \\ 3 & \text{if} & 0\le x\le 3 \\ {x}^{2} & \text{if} & x>3\end{array}$

For the following exercises, write the domain for the piecewise function in interval notation.

$f(x)=\{\begin{array}{c}x+1\,\text{if}\,x<-2 \\ -2x-3\,\text{if}\,x\ge -2\end{array}$

For the following exercises, write the domain for the piecewise function in interval notation.

$f(x)=\{\begin{array}{c}{x}^{2}-2\,\text{if}\,x<1 \\ -{x}^{2}+2\,\text{if}\,x>1\end{array}$

For the following exercises, write the domain for the piecewise function in interval notation.

$f(x)=\{\begin{array}{c}2x-3 \\ -3{x}^{2}\end{array}\,\begin{array}{c}\text{if}\,x<0 \\ \text{if}\,x\ge 2\end{array}$

Suppose the range of a function $f$ is $[-5,\,8].$ What is the range of $|f(x)|?$

Create a function in which the range is all nonnegative real numbers.

Create a function in which the domain is $x>2.$

Can the average rate of change of a function be constant?

If a function $f$ is increasing on $(a,b)$ and decreasing on $(b,c),$ then what can be said about the local extremum of $f$ on $(a,c)?$

How are the absolute maximum and minimum similar to and different from the local extrema?

For the following exercises, find the average rate of change of each function on the interval specified for real numbers $b$ or $h$ in simplest form.

$f(x)=4{x}^{2}-7$ on $[1,\,b]$

For the following exercises, find the average rate of change of each function on the interval specified for real numbers $b$ or $h$ in simplest form.

$g(x)=2{x}^{2}-9$ on $[4,\,b]$

For the following exercises, find the average rate of change of each function on the interval specified for real numbers $b$ or $h$ in simplest form.

$p(x)=3x+4$ on $[2,\,2+h]$

For the following exercises, find the average rate of change of each function on the interval specified for real numbers $b$ or $h$ in simplest form.

$k(x)=4x-2$ on $[3,\,3+h]$

For the following exercises, find the average rate of change of each function on the interval specified for real numbers $b$ or $h$ in simplest form.

$f(x)=2{x}^{2}+1$ on $[x,x+h]$

For the following exercises, find the average rate of change of each function on the interval specified for real numbers $b$ or $h$ in simplest form.

$g(x)=3{x}^{2}-2$ on $[x,x+h]$

For the following exercises, find the average rate of change of each function on the interval specified for real numbers $b$ or $h$ in simplest form.

$a(t)=\frac{1}{t+4}$ on $[9,9+h]$