Composite Functions

How to find the composite functions: examples and their solutions.

Definition

(gf)(x) = g(f(x))

It means
put [f(x)]
into g([]).

(gf)(x) is called [g of f, of x].

Example 1: (g∘f)(x)

(gf)(x) = g(f(x))

f(x) = 3x

So g(f(x)) = g(3x).

g(x) = x2 - x + 1

So g(3x) = [3x]2 - [3x] + 1.

(3x)2 = 9x2

So (given) = 9x2 - 3x + 1.

Example 2: (f∘g)(x)

(fg)(x) = f(g(x))

g(x) = x2 - x + 1

So f(g(x)) = f(x2 - x + 1).

f(x) = 3x

So f(x2 - x + 1) = 3[x2 - x + 1].

3⋅x2 = 3x2
3⋅(-x) = -3x
3⋅1 = 3

Multiplying a polynomial by a monominal

So (given) = 3x2 - 3x + 3.

In the last example,
(gf)(x) = 9x2 - 3x + 1.

And (fg)(x) = 3x2 - 3x + 3.

Then you can see that
(fg)(x) ≠ (gf)(x).

Example 3: (f∘f)(x)

(ff)(x) = f(f(x))

f(x) = 2x - 1

So f(f(x)) = f(2x - 1).

f(x) = 2x - 1

So f(2x - 1) = 2[2x - 1] - 1.

2(2x - 1) = 4x - 2

-2 - 1 = -3

So (given) = 4x - 3.

Example 4: (f∘f∘f)(x)

(fff)(x) = f(f(f(x)))

f(x) = 3x - 2

So f(f(f(x))) = f(f(3x - 2)).

f(x) = 3x - 2

So f(f(3x - 2)) = f(3(3x - 2) - 2).

3(3x - 2) = 9x - 6

-6 - 2 = -8

f(x) = 3x - 2

So f(9x - 8) = 3(9x - 8) - 2.

3(9x - 8) = 27x - 24

-24 - 2 = -26

So (given) = 27x - 26.