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

A composite function is built by substituting one function's formula into the other, 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 $(f\circ g)(x) = 3x+1$.

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

So $(g\circ f)(x) = 3x+5$.

Verification at $x=2$:

• $g(2) = 3(2)-1 = 5$, then $f(5) = 5+2 = 7$. Using the formula, $(f\circ g)(2) = 3(2)+1 = 7$ — agrees.
• $f(2) = 2+2 = 4$, then $g(4) = 3(4)-1 = 11$. Using the formula, $(g\circ f)(2) = 3(2)+5 = 11$ — agrees. …

A very common mistake is reversing the order of substitution — computing $(f\circ g)(x)$ by applying $f$ first instead of $g$ first. Remember $(f\circ g)(x) = f(g(x))$: t …