Composite functions

Given f(x) = 3x + 2, we are often asked to find f(2) or f( - 3). To do this we substitute 2 or - 3 for x. So, f(2) = 3(2) + 2 = 8 and f( - 3) = 3( - 3) + 2 =  - 7.

Sometimes, however, we are asked to find the result of a function of a function. That is, replacing x in the example above with another function.

Follow this worked example:

f(x) = 10x + 7

g(x) = 3x

Find f(g(x))

Replace x with the function

f(g(x)) = 10(g(x)) + 7

f(3x) = 10(3x) + 7

f(g(x)) = 30x + 7

Question

f(x)=x+1

g(x) = 4{x^2} + 8x - 7

Find g(f(x))

g(f(x)) = 4{(f(x))^2} + 8(f(x)) - 7

Simplify:

g(x + 1) = 4{(x + 1)^2} + 8(x + 1) - 7

g(x + 1) = 4({x^2} + 2x + 1) + 8x + 8 - 7

g(f(x)) = 4{x^2} + 8x + 4 + 8x + 1

g(f(x)) = 4{x^2} + 16x + 5

curriculum-key-fact
When you're asked to find f(g(x)) and g(f(x)), the order is important. Apart from a few special cases, f(g(x)) does not equal g(f(x)).