Functions: Question 2

Worked solution

Part (a): Finding $fg(x)$

$fg(x)$ means $f(g(x))$, apply $g$ first, then $f$.

$fg(x) = f(g(x)) = f(x^2+2)$

Substitute $x^2+2$ into $f(\cdot) = 2(\cdot) - 1$:

$fg(x) = 2(x^2+2) - 1 = 2x^2 + 4 - 1 = 2x^2 + 3$

Part (b): Finding $gf(x)$

$gf(x)$ means $g(f(x))$, apply $f$ first, then $g$: the opposite order to part (a).

$gf(x) = g(f(x)) = g(2x-1)$

Substitute $2x-1$ into $g(\cdot) = (\cdot)^2+2$:

$gf(x) = (2x-1)^2 + 2$

Expand $(2x-1)^2 = 4x^2 - 4x + 1$ carefully (keep the middle term):

$gf(x) = 4x^2 - 4x + 1 + 2 = 4x^2 - 4x + 3$

Part (c): Solving $fg(x) = gf(x)$

Set the two expressions from parts (a) and (b) equal:

$2x^2 + 3 = 4x^2 - 4x + 3$

Subtract $2x^2+3$ from both sides:

$0 = 2x^2 - 4x$

Factor (do not divide through by $x$, since that would silently lose a solution):

$0 = 2x(x - 2)$

So $x = 0$ or $x = 2$.

Check at $x=0$: $fg(0)=2(0)^2+3=3$ and $gf(0)=4(0)^2-4(0)+3=3$, equal. Check at $x=2$: $fg(2)=2(4)+3=11$ and $gf(2)=4(4)-4(2)+3=16-8+3=11$, equal.

Final answers

• (a) $fg(x) = 2x^2+3$
• (b) $gf(x) = 4x^2-4x+3$
• (c) $x = \boxed{0}$ or $x = \boxed{2}$