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)=√(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)=√(x^2+3)=f(x)`

Hence, the answers are

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