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More Lessons for Algebra, Math Worksheets

In this lesson, we will look at the composition of functions or composite functions.

**What is a Composite Function?**

A composite function is a function that depends on another function. A composite function is created when one function is substituted into another function.

For example, f(g(x)) is the composite function that is formed when g(x) is substituted for x in f(x).

f(g(x)) is read as “f of g of*x*”.

f(g(x)) can also be written as (f ο g)(*x*) or fg(*x*),

In the composition (f ο g)(*x*), the domain of f becomes g(*x*).
The following diagram shows some examples of composite functions. Scroll down the page for more examples and solutions.

**Composite Functions**

This lesson explains the concept of composite functions. An example is given demonstrating how to work algebraically with composite functions and another example involves an application that uses the composition of functions.

Examples:

1. If f(x) = x + 5 and g(x) = 3x^{2} find

(a) (f ο g)(*x*)

(b) (f ο g)(2)

(c) g(f(x))

2. A newspaper company creates routes with 50 subscribers(n) for each delivery person(d). There is a supervisor (s) for every 10 delivery persons.

(a) Write d as a function of n.

(b) Write s as a function of d.

(c) Substitute to write s as a function of n.**How to determine the value of a composite function and how to determine a composite function given two functions?**

Examples:

1. Given the functions, determine the value of each composite function.

f(x) = 2x - 1, g(x) = x^{3} - 5, h(x) = 5 - x^{2}

(a) (f ο g)(3)

(b) (g ο f)(3)

(c) (h ο g)(-1)

2. Given the functions, determine the value of each composite function.

f(x) = 4x + 1, g(x) = x^{2} - x + 5

(a) (f ο g)(x)

(b) (g ο f)(x)

**How to find the Composition of Functions?**

Example:

f(x) = x^{2} + x and g(x) = 4 - x

Find

(a) (f ο g)(*x*)

(b) (g ο f)(*x*)
**What is the composition of two functions?**

Example:

f(x) = 2x^{4} + x^{4} + 1, g(x) = √x

Find

f(g(x))

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

More Lessons for Algebra, Math Worksheets

In this lesson, we will look at the composition of functions or composite functions.

A composite function is a function that depends on another function. A composite function is created when one function is substituted into another function.

For example, f(g(x)) is the composite function that is formed when g(x) is substituted for x in f(x).

f(g(x)) is read as “f of g of

f(g(x)) can also be written as (f ο g)(

In the composition (f ο g)(

**Example:**

Given f(*x*) = *x*^{2} + 6 and g(*x*) = 2*x* – 1, find

a) (f ο g)(*x*)

b) (g ο f)(*x*)

**Solution:**

a) (f ο g)(*x*)

= f(2*x* – 1)

= (2*x* – 1)^{2} + 6

= 4*x*^{2} – 4*x* + 1 + 6

= 4*x*^{2} – 4*x* + 7

b) (g ο f)(*x*)

= g(*x*^{2} + 6)

= 2(*x*^{2} + 6) – 1

= 2*x*^{2} + 12 – 1

= 2*x*^{2} + 11

This lesson explains the concept of composite functions. An example is given demonstrating how to work algebraically with composite functions and another example involves an application that uses the composition of functions.

Examples:

1. If f(x) = x + 5 and g(x) = 3x

(a) (f ο g)(

(b) (f ο g)(2)

(c) g(f(x))

2. A newspaper company creates routes with 50 subscribers(n) for each delivery person(d). There is a supervisor (s) for every 10 delivery persons.

(a) Write d as a function of n.

(b) Write s as a function of d.

(c) Substitute to write s as a function of n.

Examples:

1. Given the functions, determine the value of each composite function.

f(x) = 2x - 1, g(x) = x

(a) (f ο g)(3)

(b) (g ο f)(3)

(c) (h ο g)(-1)

2. Given the functions, determine the value of each composite function.

f(x) = 4x + 1, g(x) = x

(a) (f ο g)(x)

(b) (g ο f)(x)

Example:

f(x) = x

Find

(a) (f ο g)(

(b) (g ο f)(

Example:

f(x) = 2x

Find

f(g(x))

Rotate to landscape screen format on a mobile phone or small tablet to use the **Mathway** widget, a free math problem solver that **answers your questions with step-by-step explanations**.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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