Question

Given () = √2 + 3, determine p(x), q(x), and r(x) such that () = ((())).

Answer 1

Start from the 'outermost' part of f(x); in other words, the square root. Set

$p(x)=\sqrt{x}$

Then, the next 'layer' is the +3, define q(x) as shown below

$q(x)=x+3$

Finally, set

$r(x)=x^2$

Thus, the composition of the three functions above is

$p(q(r(x)))=p(q(x^2))=p(x^2+3)=\sqrt{x^2+3}=f(x)$

Hence, the answers are

$\begin{gathered} r(x)=x^2 \\ q(x)=x+3 \\ p(x)=\sqrt{x} \end{gathered}$