Functions · Ordinary Level
Functions
Ordinary Level · From your notes · Tap NEXT to begin
Section 4 of 6
Composite Functions
Read it right
$fg(x) = f(g(x))$ — the inner function ($g$) runs first, the outer ($f$) second. The function touching the $x$ goes first.
Worked example — two linear rules
Given $f(x) = 3x + 1$ and $g(x) = 5x + 2$, find $fg(2)$ and $gf(5)$.
$g(2) = 5(2) + 2 = 12$, $f(12) = 3(12) + 1 = 37$ $\Rightarrow fg(2) = 37$
$f(5) = 3(5) + 1 = 16$, $g(16) = 5(16) + 2 = 82$ $\Rightarrow gf(5) = 82$
$fg(2) = 37,\ gf(5) = 82$
Order matters
$fg(2)$ and $gf(5)$ gave different numbers. $fg$ is not the same function as $gf$.
You try
Given $g(x) = x^2 + 3x$ and $f(x) = 5x - 1$, find $fg(-3)$ and $gf(-2)$.
Inner first. For $fg(-3)$: work out $g(-3)$, then put it into $f$. Mind $(-3)^2 = 9$.
$g(-3) = (-3)^2 + 3(-3) = 0$, $f(0) = -1$ $\Rightarrow fg(-3) = -1$
$f(-2) = 5(-2) - 1 = -11$, $g(-11) = (-11)^2 + 3(-11) = 88$ $\Rightarrow gf(-2) = 88$
$fg(-3) = -1,\ gf(-2) = 88$
Algebraic composites — answer in terms of $x$
Keep it in brackets
Wherever the outer function has an $x$, replace it with the whole inner expression — in brackets. Multiply out and tidy.
Worked example — outer linear, inner linear
Given $f(x) = 3x + 2$ and $g(x) = 5x + 1$, find $fg(x)$.
$fg(x) = f(5x + 1) = 3(5x + 1) + 2$
$= 15x + 3 + 2 = 15x + 5$
$fg(x) = 15x + 5$
Worked example — same functions, both ways
Given $f(x) = 3x + 1$ and $g(x) = 5x - 2$, find $fg(x)$ and $gf(x)$.
$fg(x) = f(5x - 2) = 3(5x - 2) + 1 = 15x - 5$
$gf(x) = g(3x + 1) = 5(3x + 1) - 2 = 15x + 3$
$fg(x) = 15x - 5,\ gf(x) = 15x + 3$
You try
Given $f(x) = x^2 + 1$ and $g(x) = 2x - 3$, find $fg(x)$.
$fg(x) = f(2x - 3) = (2x - 3)^2 + 1$. Remember $(2x-3)^2$ means $(2x-3)(2x-3)$.
$(2x - 3)^2 + 1 = (2x - 3)(2x - 3) + 1$
$= 4x^2 - 12x + 9 + 1 = 4x^2 - 12x + 10$
$fg(x) = 4x^2 - 12x + 10$
Part 4 done.
Next up: Graphing Linear & Quadratic. Head back to the hub for the next part.