Composite Functions – An Ultimate Guides On How To Solve Composite Functions With 5 Examples
What is a Composite Function?
How to Solve Composite Functions?
To indicate the composition of a function, we utilize a small circle symbol (∘). Below are the steps outlining how to solve a composite function.
For example
(f ∘ g) (x) = f [g (x)]
(f ∘ g) (x) = f [g (x)]
(f ∘ g) (x²) = f [g (x²)]
The first step is to replace the variable x in the outer function with the inner function. Afterward, simplify the resulting function.
The sequence or order of functions in the composition is significant because (f ∘ g) (x) is not equivalent to (g ∘ f) (x).
Examples of Problem Solved
example 1
Given the functions f (x) = x2 + 4 and g (x) = 2x – 1, find (f ∘ g) (x).
solving
Substitute x with 2x – 1 in the function f(x) = x2 + 4
(f ∘ g) (x) = (2x – 1)2 + 4 = (2x – 1) (2x – 1) + 4
Apply FOIL
= 4x2 – 4x + 1 + 4
= 4x2 – 4x + 5
Example 2
Given f (x) = 2x + 2, find (f ∘ f) (x).
Solution
(f ∘ f) (x) = f[f(x)]
= 2(2x + 2) + 2
= 4x + 4
Example 3
Find (g ∘ f) (x) given that, f (x) = 2x + 3 and g (x) = –x2 + 5
Solving
⟹ (g ∘ f) (x) = g [f (x)]
Replace x in g(x) = –x2 + 5 with 2x + 3
= – (2x + 3)2 + 5
= – (4x2 + 12x + 9) + 5
= –4x2 – 12x – 9 + 5
= –4x2 – 12x – 4
Example 4
Find (g ∘ f) (x) if, f(x) = 6 x² and g(x) = 14x + 4
Solving
⟹ (g ∘ f) (x) = g [f(x)]
Substitute x in g(x) = 14x + 4 with 6 x²
⟹g [f(x)] =14 (6 x²) + 4
= 84 x² + 4
Example 5
Calculate (f ∘ g) (x) using f(x) = 2x + 3 and g(x) = -x 2 + 1,
Solving
(f ∘ g) (x) = f(g(x))
= 2 (g(x)) + 3
= 2(-x 2 + 1) + 3
= – 2 x 2 + 5