함수와 그래프 전체 문제

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

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

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

$f(x)=\{\begin{array}{cc}1 & \text{if }x\le -3 \\ 0 & \text{if }x>-3\end{array}$

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.

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

Graph $y=\frac{1}{{x}^{2}}$ on the viewing window $[-0.5,-0.1]$ and $[0.1,0.5].$ Determine the corresponding range for the viewing window. Show the graphs.

Graph $y=\frac{1}{x}$ on the viewing window $[-0.5,-0.1]$ and $[0.1,\,0.5].$ Determine the corresponding range for the viewing window. Show the graphs.

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 $x>2.$

The height $h$ of a projectile is a function of the time $t$ it is in the air. The height in feet for $t$ seconds is given by the function $h(t)=-16{t}^{2}+96t.$ What is the domain of the function? What does the domain mean in the context of the problem?

The cost in dollars of making $x$ items is given by the function $C(x)=10x+500.$

• ⓐThe fixed cost is determined when zero items are produced. Find the fixed cost for this item.
• ⓑWhat is the cost of making 25 items?
• ⓒSuppose the maximum cost allowed is $1500. What are the domain and range of the cost function, $C(x)?$

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?

How does the graph of the absolute value function compare to the graph of the quadratic function, $y={x}^{2},$ in terms of increasing and decreasing intervals?