Chapter 1.2, Problem 5E

(a)

To determine

To find:The domain and range of the function $y=4-x^2$ .

Answer to Problem 5E

The domain and range of the function $y=4-x^2$ is $\left(-\infty ,\infty \right)$ and $\left(-\infty ,4\right]$ respectively.

Explanation of Solution

Given information:The given function is $y=4-x^2$ .

Calculation:

The function $y=4-x^2$ is defined for all the real values of $x$ . Domain is the set of all input values. The domain of the function is the set of real numbers or the interval $\left(-\infty ,\infty \right)$ .

The function $x^2$ is always positive for all the real values of $x$ .

  $x^2>0$

Multiply both sides by $-1$ .

  $-x^2<0$

Add $4$ to both sides.

  $\begin{array}{c}4-x^2<4\\ y<4\end{array}$

So, the range of the function $y=4-x^2$ is the interval $\left(-\infty ,4\right]$ .

Therefore, the domain and range of the function $y=4-x^2$ is $\left(-\infty ,\infty \right)$ and $\left(-\infty ,4\right]$ respectively.

(b)

To determine

To graph:The function $y=4-x^2$ .

Explanation of Solution

Given information:The function is $y=4-x^2$ .

Graph:

The graph of the function $y=4-x^2$ is drawn below:

  

Figure(1)

Interpretation:The graph of the function $y=4-x^2$ is an downwards parabola with vertex at the point $\left(0,4\right)$ .