Q.If $f(x)=2x+1$ and $g(x)=x^2-3$, find $(f\circ g)(x)$ and $(g\circ f)(x)$, and verify both at $x=2$.

A composite is built by substituting the whole inner rule into the outer rule, applying the inner function first.

Finding $(f\circ g)(x)$: by definition $(f\circ g)(x)=f(g(x))$, so $g$ is applied first.

So $(f\circ g)(x)=2x^2-5$.

Finding $(g\circ f)(x)$: here $f$ is applied first.

So $(g\circ f)(x)=4x^2+4x-2$.

Verification at $x=2$ (dual check):

• $g(2)=2^2-3=1$, then $f(1)=2(1)+1=3$. Formula: $(f\circ g)(2)=2(2)^2-5=8-5=3$. ✓
• $f(2)=2(2)+1=5$, then $g(5)=5^2-3=22$. Formula: $(g\circ f)(2)=4(2)^2+4(2)-2=16+8-2=22$. ✓ …

The commonest error is reversing the order — computing $(f\circ g)$ by applying $f$ first. Remember $(f\circ g)(x)=f(g(x))$: the function nearer $x$ (on the righ …