GB-41-02-001 · Composite functions

Functions are defined by

f(x)=8x+4

and

g(x)=x^2+8

. Here fg(x) means f(g(x)).

(a) Find fg(8).

(b) Find gf(x), giving your answer in expanded form.

Part (a) Composite Functions Booklet p. 30

Method

1) Always work out from the inside bracket!

2) For gf(x), substitute f(x) into g( )

3) Expand using 2×2 grid and simplify

Worked Example (GB-41-02-001 · Part (a))

Find fg(8).

Booklet p. 30 — Always work out from the inside bracket!

fg(8) = f\big(g(8)\big)

① Inside first:

g(8) = (8)^2 + 8 = 64 + 8 = \mathbf{72}

② Put the answer into

f

f(72) = 8 \times 72 + 4 = 576 + 4

580

Part (b) Composite Functions Booklet p. 30

Method

1) Always work out from the inside bracket!

2) For gf(x), substitute f(x) into g( )

3) Expand using 2×2 grid and simplify

Worked Example (GB-41-02-001 · Part (b))

Find gf(x), giving your answer in expanded form.

Booklet p. 30 — Always work out from the inside bracket! Substitute

f(x) = 8x + 4

into

g(x)

gf(x) = g(8x + 4) = (8x + 4)^2 + 8

Expand

(8x + 4)(8x + 4)

using 2×2 Grid (Booklet p. 2):

×	$8x$	$+4$
$8x$	$64x^2$	$+32x$
$+4$	$+32x$	$+16$

Add

+8

\quad 64x^2 + 64x + 16 + 8 = \mathbf{64x^2 + 64x + 24}

64x^{2} + 64x + 24