[MusicalityGuide] Additional Mathematics Form 4 Chapter 1 (Functions): How to do Composite Functions

Finding g(x) when f(x) and fg(x) are given:

Given f(x) = 2x + 1 and fg(x) = 6x + 5, find g(x)

fg(x)        = 6x + 5
2g(x) + 1 = 6x + 5
2g(x)       = 6x + 5 - 1
g(x)         = (6x + 4)/2
g(x)         = 3x + 2

Finding f(x) when g(x) and fg(x) are given:

Given g(x) = 3x - 9 and fg(x) = 6x - 11, find f(x)

fg(x)            = 6x - 11
f(3x - 9)       = 6x - 11

For this one, find the inverse in g(x). It is one of the easiest ways to solve this question.

g(x)         = y
3x - 9      = y
3x           = y + 9
x             = (y + 9)/3
g-1(x)     = (x + 9)/3

Why do I ask to find the inverse? If you combine g(x) with g-1(x), it combines to get an identity function. What are identity functions used for? If you combine an identity function, I with any function, f(x), it will get that function back, f(x).

fg[g-1(x)]     = 6[(x+9)/3] - 11
f(x)               = 2(x + 9) - 11
f(x)               = 2x + 18 - 11
f(x)               = 2x + 7

See, easy as that!