Question

Find the domain and range of the real function $f\left(x\right)=\sqrt{25-x^2}$.

Answer 1

$f\left(x\right)=\sqrt{25-x^2}$

For the function to be defined, the expression inside the root must be greater than to equal to $0$.

$25-x^2\ge 0$

$x^2\le 25$

$|x|\le 5$

$-5\le x\le 5$

The domain of

$f\left(x\right)=\left[-5,5\right]$

When $x=\pm 5,f\left(x\right)=0$

When $x=0,f\left(x\right)=5$, this is maximum value.

The range is $R=[0,5]$