Question

Express the following function, as a composition of two functions f and g. h(x)= x^2/(x^+4)

Answer 1

According to defintion of compostion of functions, we can get:

$(f\circ g)(x)=f(g(x))$

But we know, that:

$\displaystyle h(x)= \frac{x^2}{x^2+4}$

Our goal, is to express that function as a composition:

$(f\circ g)(x)=h(x) \Rightarrow f(g(x))=h(x)$

I.e:

$\displaystyle f(g(x))=\frac{x^2}{x^2+4}$

If $\displaystyle f(y)= \frac{y}{y+4}$ then $g(x)=x^2$. Hence, $\displaystyle h(x)=(f\circ g)(x)=f(g(x))= \frac{g(x)}{g(x)+4} \Rightarrow f(g(x))=f(x^2)= \frac{x^2}{x^2+4}$