**Composite Functions – An Ultimate Guides On How To Solve Composite Functions With 5 Examples**

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In mathematics, a function can be defined as a principle that establishes a connection between a specific group of inputs and a corresponding range of potential outputs. It is essential to emphasize that each input within a function is uniquely associated with a single output.

The act of assigning names to functions is referred to as function notation. Common symbols used in function notation include “f(x) = …,” “g(x) = …,” “h(x) = …,” and so on.

This text aims to provide instruction on composite functions and their solution methods.

**What is a Composite Function?**

A composite function is typically a function that is embedded within another function. The process of composing a function involves substituting one function with another.

Given two functions, creating a new function by combining them through composition is possible. The steps involved in this process resemble those used to solve any function for a given value. These functions are commonly referred to as composite functions.

For instance, f [g (x)] represents the composite function of f (x) and g (x). The notation f [g (x)] is read as “f of g of x.” The function g (x) is called the inner function, while the function f (x) is called the outer function. Consequently, f [g (x)] can also be interpreted as “the function g is the inner function of the outer function f.”

**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) = x^{2} + 4 and g (x) = 2x – 1, find (f ∘ g) (x).

**solving**

Substitute x with 2x – 1 in the function f(x) = x^{2} + 4

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

Apply FOIL

= 4x^{2} – 4x + 1 + 4

= 4x^{2} – 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) = –x^{2} + 5

**Solving**

⟹ (g ∘ f) (x) = g [f (x)]

Replace x in g(x) = –x^{2} + 5 with 2x + 3

= – (2x + 3)^{2} + 5

= – (4x^{2} + 12x + 9) + 5

= –4x^{2} – 12x – 9 + 5

= –4x^{2} – 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