Chapter 3.7, Problem 4GP

To determine

To sketch: the graph of the function and then intercept given piece of the function.

Answer to Problem 4GP

The final range of the function is $\left[0,\infty \right)$ .

Explanation of Solution

Given information:

The function $f\left(x\right)=\{\begin{array}{l}8x\text{if}0\le x\le 40\\ 12x-160\text{if}x>40\end{array}$ Models J’s earnings in dollars in a week when he works $x$ hours.

Calculation:

Graph:

  

Given the $0\le x\le 40$ for the first equation $8x$ .

For the second equation the $12x-160$ .

So these to inequalities $0\le x\le 40$ and $x>40$ .

Therefore, the domain of the function is $\left[0,40\right]$

The range of the first function $f\left(x\right)=8x$ when $x=40$ boundary value 320 is not included.

Therefore, the range of the first function is $\left[0,320\right]$ .

For the second equation $f\left(x\right)=12x-160$ when $x=40$ (not included), $f\left(x\right)=320$ (not included)

The domain of the function $\left(40,\infty \right)$

Therefore, the range of the second function starts from $\left(320,\infty \right)$

Therefore, the combined domain of the total function is $\left[0,\infty \right)$ .

So the final range of the function is $\left[0,\infty \right)$ .