Question

Hello, I am having such a hard time solving for restrictions of functions. Could you please show me the steps to solving number 13?

Answer 1

Lets gfind the composite function first:

$\begin{gathered} s(q(x))=s(5\sqrt{x}) \\ =\frac{1}{5}(5\sqrt{x})^2 \\ =\frac{25x}{5} \\ =5x \end{gathered}$

Therefoe , we have:

$s(q(x))=5x$

Now, to find the domain we need to remember that the domain of the function s(q(x)) is the set of numbers x in the domain of q for which q(x) is in the domain of s. To find it we will make the following steps:

• Find the doain of sq.

,

• Find the domain of s.

,

• Find the values x in the domain of q for which q(x) is in the domain of f; that is, we will exclude the values, x, form the domain of q for which q(x) is not in the domain of f.

We know that the functio q is tdefinesd as:

$q(x)=5\sqrt{x}$

and we know that, in the real numbers, the square root for negative numbers is not defied which means that x has to be a non-*egative number in order for sqto be defined, tahat is:

$dom\text{ }q=\lbrack0,\infty)$

Now, function s is defined as:

$s(x)=\frac{1}{5}x^2$

and since this is a polynomial, its domain is all the real number set, that is, the x value for function s can be any number. Then w, don't need to exclude any value of x for function s.

Therefore, the domain of the composite function s(q(x)) is:

$\lbrack0,\infty)$