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Here we will learn about composite functions including how to evaluate composite functions and how to solve problems involving composite functions.

There are also composite functions* *worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

**Composite functions** are when the output of one function is used as the input of another.

If we have a function f and another function g , the function fg(x) ,

said as “ f of g of x ”, or “ fg of x ”, is the composition of the two functions.

The order of how the functions are applied is important.

We are finding a function of a function, and if we have two or more functions, there could be many different permutations of those functions leading to many different composite functions.

fg(x) could be written as f[g(x)] which shows that the inner function must be applied before the outer function. We can evaluate composite functions for numerical values or find the algebraic expression for the new function.

To evaluate a composition of functions for a numerical value we can just substitute the value into the inside function and then use the result of that function to substitute into the outside function.

E.g.

Here the function h is described by h(x)=x^2 and the function f by f(x)=x-5 .

We can find hf(2) by finding f(2) , which gives us −3 ,

and then finding h(−3) which gives us 9.

If we want to find the expression for the function fh(x) , we can replace the x in the expression for f(x) with the expression for h(x) .

There are other types of notation that can be used for composite functions.

E.g.

The composition of functions f and g can be written using a small circle,

(f ∘ g)(x)=fg(x) .

In A level mathematics we look at composite functions in more depth by finding the derivatives of composite functions using a process called the chain rule. The derivative of a function gives us an expression for the function’s gradient at any point.

In order to evaluate composite functions:

**Use the****number to be evaluated as the input for the inner function and substitute it into the expression.****Find the output for the inner function and substitute it into the expression for the outer function.****Find the output for the outer function and repeat the process if there are any further outer functions.**

Get your free composite functions worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEGet your free composite functions worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEThis lesson is part of the larger topic, functions in algebra. It may be helpful to take a look at the summary page, functions in algebra, before moving on to the related lessons in more detail:

If f(x)=4x and g(x)=x^2-1 , find fg(4) :

**Use the number to be evaluated as the input for the inner function and substitute it into the expression**.

2**Find the output for the inner function and substitute it into the expression for the outer function.**

3**Find the output for the outer function and repeat the process if there are any further outer functions.**

If f(x)=2x+1 find ff(3) :

**Use the number to be evaluated as the input for the inner function and substitute it into the expression**.

f\left( 3 \right)=2\left( 3 \right)+1

**Find the output for the inner function and substitute it into the expression for the outer function.**

\begin{aligned}
& f\left( 3 \right)=7 \\\\
& f\left( 7 \right)=2\left( 7 \right)+1
\end{aligned}

f(7)=15, \text{ so } ff(3)=15.

If f(x)=3x+2, g(x)=x^{2} and h(x)=\frac{x}{2}, find hfg(-5) :

**Use the number to be evaluated as the input for the inner function and substitute it into the expression**.

g\left( -5 \right)={{\left( -5 \right)}^{2}}

**Find the output for the inner function and substitute it into the expression for the outer function.**

\begin{aligned}
& g\left( -5 \right)=25 \\\\
& f\left( 25 \right)=3\left( 25 \right)+2
\end{aligned}

f(25)=77

Repeat Step 2 for h(77)

h(77)=\frac{77}{2}=38.5

So hfg(-5)=38.5

In order to find composite functions:

**Take the most inner function and substitute in to the next outer function wherever there is an x .****Simplify the expression as appropriate.****Repeat for any further outer functions.**

If f(x)=3x-1 and g(x)=x^2+2, find fg(x) :

** Take the most inner function and substitute in to the next outer function wherever there is an ** x.

\begin{aligned} & fg\left( x \right)= f({{x}^{2}}+2) \\\\ & =3({{x}^{2}}+2)-1 \end{aligned}

**Simplify the expression as appropriate.**

\begin{aligned}
& fg\left( x \right)=3{{x}^{2}}+6-1 \\\\
& =3{{x}^{2}}+5
\end{aligned}

**Repeat for any further outer functions.**

No further outer functions, so fg(x)=3x^2+5 .

If f(x)=2x+1 and g(x)=x^2, find gf(x) :

** Take the most inner function and substitute in to the next outer function wherever there is an ** x.

\begin{aligned}
& gf\left( x \right)=g\left( 2x+1 \right) \\\\
& ={{\left( 2x+1 \right)}^{2}}
\end{aligned}

**Simplify the expression as appropriate.**

\begin{aligned}
& gf\left( x \right)=4{{x}^{2}}+2x+2x+1 \\\\
& =4{{x}^{2}}+4x+1
\end{aligned}

** Repeat for any further outer functions.**

No further outer functions, so gf(x)=4x^2+4x+1.

If f(x)=x+2, g(x)=x^2+1 and h(x)=3x, find fgh(x):

** Take the most inner function and substitute in to the next outer function wherever there is an ** x.

\begin{aligned}
& fgh\left( x \right)=fg\left( 3x \right) \\\\
& =f\left[ {{\left( 3x \right)}^{2}}+1 \right]
\end{aligned}

**Simplify the expression as appropriate.**

fgh(x)=f(9x^2+1)

**Repeat for any further outer functions.**

\begin{aligned} & fgh\left( x \right)=f\left( 9{{x}^{2}}+1 \right) \\\\ & =\left( 9{{x}^{2}}+1 \right)+2 \\\\ & =9{{x}^{2}}+3 \end{aligned}

So, fgh(x)=9x^2+3 .

**Mistaking composite functions for the product of the functions**

A common mistake is to think that fg(x) means f(x) \times g(x) .

E.g.

If f(x)=x+1 and g(x)=2x-3 , the error will be to think that

fg(x)=(x+1)(2x-3) , rather than the correct answer of fg(x)=2x-2 .

**The functions are applied from left to right instead of right to left**

A common error is to apply the functions in the wrong order.

fgh(x) means we should apply the functions in the order h , then g , then f , which is right to left.

A common mistake is to apply them from left to right. Thinking of fgh(x) as f[g(h(x))] can help to use the functions in the correct order, i.e. starting with the innermost function and working outwards.

1. Evaluate gf(2) when f(x)=3x+5 and g(x)=2x-1

14

33

3

21

f(2)=11 , then g(11)=21.

2. Evaluate fg(-3) when f(x)=5x-2 and g(x)=x^2+1

290

48

-42

-52

g(-3)=10 , then f(10)=48.

3. Evaluate ghf(4) when f(x)=x-3, g(x)=x^2 and h(x)=5x

25

77

320

5

f(4)=1 , then h(1)=5, then g(5)=25.

4. Find gf(x) when f(x)=4x+6 and g(x)=\frac{x}{2}-3

2x-6

(\frac{x}{2}-3)(4x+6)

2x

2x+3

gf(x)=g(4x+6)=\frac{4x+6}{2}-3=2x.

5. Find fg(x) when f(x)=x^2-1 and g(x)=2x+3

2x^2+1

4x^2+12x+8

(x^2-1)(2x+3)

4x^2+12x+10

fg(x)=f(2x+3)=(2x+3)^2-1.

6. Find hgf(x) when f(x)=x+1, g(x)=x^2 and h(x)=7x

7x^4+7x^3

49x^2+1

7x^2+1

7x^2+14x+7

hgf(x)=7(x+1)^2.

1. f(x)=3x+1 and g(x)=x^2+2 .

(a) Find gf(-2)

(b) Find an expression for fg(x).

**(4 marks)**

Show answer

(a)

f(-2)=-5

**(1)**

g(-5)=27

**(1)**

(b)

g(x) substituted into f, 3(x^2+2)+1

**(1)**

3x^2+7

**(1)**

2. f(x)=2x^2-1, g(x)=x+6, h(x)=3(x-4)

(a) Find ghg(3)

(b) Write an expression for hfg(x).

**(6 marks)**

Show answer

(a)

hg(3)=15

**(1)**

ghg(3)=21

**(1)**

(b)

g correctly substituted into f, fg(x)=2(x+6)^2-1

**(1)**

fg(x) simplified , fg(x)=2x^2+24x+71

**(1)**

fg correctly substituted into h

**(1)**

Fully simplified answer hfg(x)=6x^2+72x+201

**(1)**

3. f(x)=2x+1 and g(x)=x^2-2

(a) Find fg(x)

(b) Find when fg(x)=gf(x)

**(6 marks)**

Show answer

(a)

2(x^2-2)+1

**(1)**

2x^2-3

**(1)**

(b)

gf(x)=(2x+1)^2-2

**(1)**

Set equal 2x^2-3=4x^2+4x-1

**(1)**

Attempt to solve/ factorise

2(x+1)(x+1)=0

**(1)**

**(1)**

You have now learned how to:

- Where appropriate, interpret simple expressions as functions with inputs and outputs
- Interpret the succession of two functions as a ‘composite function’

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