Composite functions

Why fg(x) means "g first, then f", how to chain outputs into inputs for numbers and for algebra, and why the order almost always matters.

Algebra · Functions

Why fg(x) means "g first, then f", how to chain outputs into inputs for numbers and for algebra, and why the order almost always matters.

Why it works

fg(x)

chains two machines: it means

f\big(g(x)\big)

— feed $x$ into $g$ , then feed $g$ 's output into $f$ . Read it inside-out: the function written CLOSEST to

x

acts first. (It is not

f \times g

— no multiplying happens between the machines.)

With numbers: two small steps.

f(x) = 3x - 4

g(x) = x^2 + 1

fg(2): \quad g(2) = 5, \;\text{then}\; f(5) = 11.

gf(2): \quad f(2) = 2, \;\text{then}\; g(2) = 5.

Different answers — order matters.

fg

and

gf

agree only by coincidence, so never swap them to "save time".

With algebra: substitute the whole inner formula.

fg(x)

replaces every

x

f

's formula with the ENTIRE expression for

g(x)

, brackets intact:

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

gf(x) = (3x - 4)^2 + 1 = 9x^2 - 24x + 17

— note the full bracket-square in

gf

, middle term and all. The number checks agree:

fg(2) = 3(4) - 1 = 11

. ✓

Solving with composites: build, then solve. "Solve

fg(x) = 26

": first build

fg(x) = 3x^2 - 1

, then solve

3x^2 - 1 = 26

—

x^2 = 9

x = \pm 3

. The composite is just a new machine; the old equation skills finish the job.

A function composed with itself: $ff(x)$ . Same chaining:

ff(x) = 3(3x - 4) - 4 = 9x - 16

. Nothing new — the inner machine just happens to be the same one.