Chapter 9.1, Problem 21P

Identify the domain and range of each function.

$y=\frac{3}{4}{x}^2-2$

To determine

To find the domain and range of the given function.

Answer to Problem 21P

The domain and range of $y=\frac{3}{5}{x}^2-2$ is given by $-\infty <x<\infty$ and $y\ge -2$ .

Explanation of Solution

Given information:

The given function is $y=\frac{3}{5}{x}^2-2$ .

Consider the given function $y=\frac{3}{5}{x}^2-2$ .

The function is of the form $y=a{x}^2+c$ . Hence, the domain of $y=\frac{3}{5}{x}^2-2$ 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 $a=\frac{3}{5}$ , hence the range is $f\left(x\right)\ge -2\Rightarrow y\ge -2$ .

Therefore, the domain and range of $y=\frac{3}{5}{x}^2-2$ is given by $-\infty <x<\infty$ and $y\ge -2$ .