Composite Function

How to solve the composite function: definition, 4 examples, and their solutions.

Definition

Definition

(g ∘ f)(x) = g(f(x))

To find g(f(x)),
put f(x)
into g(  ).

Function

Example 1

Example

Solution

(g ∘ f)(x) = g(f(x))

Solve this from the inside.

f(x) = 3x

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

g(x) = x2 - x - 1

To solve g(3x),
put [3x]
into g(  ) = [  ]2 - [  ] - 1..

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

[3x]2
= 32x2
= 9x2

Power of a Product

So (g ∘ f)(x) = 9x2 - 3x - 1.

Example 2

Example

Solution

(f ∘ g)(x) = f(g(x))

Solve this from the inside.

g(x) = x2 - x - 1

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

f(x) = 3x

To solve f(x2 - x - 1),
put [x2 - x - 1]
into f(  ) = 3[  ].

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

3[x2 - x - 1]
= 3x2 - 3x - 3

Multiply a Monomial and a Polynomial

So (f ∘ g)(x) = 3x2 - 3x - 3.

Compare (f ∘ g)(x) = 3x2 - 3x - 3
and the previous answer (g ∘ f)(x) = 9x2 - 3x - 1.

As you can see,
(f ∘ g)(x) ≠ (g ∘ f)(x).

Example 3

Example

Solution

(f ∘ f)(x) = f(f(x))

Solve this from the inside.

f(x) = 2x - 1

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

f(x) = 2x - 1

To solve f(2x - 1),
put [2x - 1]
into f(  ) = 2[  ] - 1.

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

2[2x - 1] = 4x - 2

-2 - 1 = -3

So (f ∘ f)(x) = 4x - 3.

Example 4

Example

Solution

(f ∘ f ∘ f)(x) = f(f(f(x)))

Solve this from the inside.

f(x) = 3x + 1

So f(f(f(x))) = f(f(3x + 1)).

f(x) = 3x + 1

To solve f(3x + 1),
put [3x + 1]
into f(f(  )) = f(3[  ] + 1).

Then f(f(3x + 1)) = f(3[3x + 1] + 1).

3[3x + 1] = 9x + 3

+3 + 1 = +4

f(x) = 3x + 1

To solve f(9x + 4),
put [9x + 4]
into f(  ) = 3[  ] + 1.

Then f(9x + 4) = 3[9x + 4] + 1.

3[9x + 4] = 27x + 12

+12 + 1 = +13

So (f ∘ f ∘ f)(x) = 27x + 13.