Question

Find the domain and range of $f\left(x\right)=\sqrt{(x-1)(3-x)}$

Answer 1

We have,

$f\left(x\right)=\sqrt{(x-1)(3-x)}$

Now,

Under square root the polynomial must be positive. The domain is obtained by solving:-

$(x-1)(3-x)\ge 0$

You can obtain the zeroes of the polynomial

$(x-1)(3-x)=0$

If

$x-1=0$

$x=1$

If

$3-x=0$

$\Rightarrow x=3$

So,

$x=1,3$

So the polynomial is solved in the external intervals.

$x>1,x>3$

So,

$x\in (1,+\infty )$

Since the function $f\left(x\right)$ is positive, due to result of square root,

The range includes all positive real numbers $x\ge 0$

Hence, this is the answer.