Chapter 11, Problem 8CA

To determine

To find: The numbers that are in the range of the function.

Answer to Problem 8CA

The required numbers that are in range of the function are,

  

Explanation of Solution

Given information:

The function: $y=\{\begin{array}{l}{x}^2+4x+7,\text{if}x\le -1\\ \frac{1}{2}x+2,\text{if}x>-1\end{array}$

The numbers:

  

The given function are $y=x^2+4x+7,\text{if}x\le -1$ and $y=\frac{1}{2}x+2,\text{if}x>-1$ .

Substitute $-1$ for $x$ in the function $y=x^2+4x+7$ .

  $\begin{array}{l}y={\left(-1\right)}^2+4\left(-1\right)+7\\ y=1-4+7\\ y=-3+7\\ y=4\end{array}$

It is seen that the parabola is open upward.

This means the range is $y\ge 4$ .

Substitute $-1$ for $x$ in the function $y=\frac{1}{2}x+2$ .

  $\begin{array}{l}y=\frac{1}{2}\left(-1\right)+2\\ y=-\frac{1}{2}+2\\ y=\frac{3}{2}\end{array}$

It is seen that the line is increasing from left to right.

This means the range is $y>\frac{3}{2}$ .

Hence, the required numbers that are in range of the function are,