Chapter 1, Problem 6RE

To determine

To find: The domain and the range of the function.

Answer to Problem 6RE

Solution:

The domain of the function g(x) is $\left[-2,2\right]$.

The range of the function g(x) is $\left[0,4\right]$.

Explanation of Solution

Given:

The function $g\left(x\right)=\sqrt{16-x^4}$.

Definition used:

The domain is the set of all input values of the function for which the function is real and defined.

Calculation:

Consider the expression $16-x^4$.

Since the expression inside the radical cannot be negative, $16-x^4\ge 0$.

Thus, the undefined points are obtained follows:

$\begin{array}{l}16\ge x^4\\ -16\le x^4\le 16\\ -2\le x\le 2\end{array}$

Thus, the solutions are, $x=-2$ and $x=2$.

Hence, the function is undefined for $x=-2$ and $x=2$.

Therefore, the domain of the function is $\left[-2,2\right]$.

The range is the set of all output values of the function. So, obtain the output for the input values lies in the interval $\left[0,2\right]$.

If $x=-2$, then $g\left(x\right)=0$.

If $x=0$, then $g\left(x\right)=4$.

If $x\ne 0$, then $g\left(x\right)\le 4$.

If $x=2$, then $g\left(x\right)=0$.

Thus, it is observed that the range of the function is $\left[0,4\right]$.