Find the fuction $g$.

Question

If $f:x \mapsto x^2 + 3$, find function $g$ such that $gf:x \mapsto 2x^2 + 3$.

I don't know how to do it, there is no such example in my book.

Help?

Answer 1

$$f(x)=x^2+3,\quad g(f(x))=2x^2+3\\ \implies g(x^2+3)=2x^2+3=2(x^2+3)-3\\ \implies g(x)\text{ could be }g(x):=2x-3$$ If $gf=g(x)\times f(x)$, then $$g(x)\times f(x)=2x^2+3\\ \implies g(x)=\dfrac{2x^2+3}{x^2+3}=1+\dfrac{x^2}{x^2+3}$$