Chapter 9.1, Problem 20P

Identify the domain and range of each function.

$y=-\frac{1}{2}{x}^2+3$

To determine

To find the domain and range of the given function.

Answer to Problem 20P

The domain and range of $y=-\frac{1}{2}{x}^2+3$ is given by $-\infty <x<\infty$ and $y\le 3$ .

Explanation of Solution

Given information:

The given function is $y=-\frac{1}{2}{x}^2+3$ .

Consider the given function $y=-\frac{1}{2}{x}^2+3$ .

The function is of the form $y=a{x}^2+c$ . Hence, the domain of $y=-\frac{1}{2}{x}^2+3$ is given by $-\infty <x<\infty$ and in interval notation it can be expressed as $\left(-\infty ,\infty \right)$ .

For a parabola $y=a{x}^2+bx+c$ with vertex $\left(x_v,y_v\right)$ ,

If $a<0$ the range is $f\left(x\right)\le y_v$ ,

If $a>0$ the range is $f\left(x\right)\ge y_v$ ,

Here $a=-\frac{1}{2}$ , hence the range is $f\left(x\right)\le 3\Rightarrow y\le 3$ .

Therefore, the domain and range of $y=-\frac{1}{2}{x}^2+3$ is given by $-\infty <x<\infty$ and $y\le 3$ .