**Functions**

In maths, a **function** is something that takes an input and produces an output. Functions may be given in the form of function machines – or they may be given as mathematical expressions.

Make sure you are happy with the following topics before continuing.

## Test your skills with online exams on the MME Revision Platform

##### 5 Question Types

Our platform contains 5 question types: simple, multiple choice, multiple answers, fraction and image based questions. More question types are coming soon.

##### Video Solutions

Premium users have access to Video Solutions for every single exam question. Our expert Maths tutors explain all parts of the question and answer in detail. Follow along and improve your grades.

##### Written Solutions

Get written solutions for every single exam question, detailing exactly how to approach and answer each one, no matter the difficulty or topic.

##### Track your progress

Every exam attempt is stored against your unique student profile, meaning you can view all previous exam and question attempts to track your progress over time.

## Skill 1: Evaluating Functions

Evaluating functions involves putting numbers into the function to get the result.

**Example: **A function is given by f(x) = 3x+1, Find f(10)

All this requires is to replace x with 10 and calculate the result.

When we input **10** into this** function** that would look like:

f(\textcolor{red}{10}) = 3\times \textcolor{red}{10} + 1 = 31.

## Type 2: Composite Functions

A **composite function** is the result of one function being applied immediately after the other.

**Example:** Let f(x)=\textcolor{red}{2x-3} and g(x)=\textcolor{blue}{x+1}, find fg(x)

To find fg(x) we replace x in f(x) with g(x)

fg(x) = f(g(x)) = \textcolor{red}{2(}\textcolor{blue}{x+1}\textcolor{red}{) - 3}

Next we can expand the brackets and simplify if required.

\textcolor{red}{2(}\textcolor{blue}{x+1}\textcolor{red}{) - 3} = 2x+2-3 = 2x-1

## Type 3: Inverse Functions

An **inverse function** is a function acting in reverse. The **inverse function** of f(x) is given by f^{-1}(x), and it tells us how to go from an output of f(x) back to its input.

**Example:** Given that f(x) = \dfrac{x+8}{3}, find f^{-1}(x)

**Step 1:** Write the equation in the form x = f(y)

For this we need to replace all the x‘s in the equation with y‘s and set the equation equal to x

f(x) = \dfrac{x+8}{3} becomes x= \dfrac{y+8}{3}

**Step 2:** Rearrange the equation to make y the subject.

\begin{aligned}x&= \dfrac{y+8}{3} \\ 3x& = y+8 \\ 3x-8 &= y \end{aligned}

**Step 3:** Replace y with f^{-1}(x)

\begin{aligned}y & = 3x-8 \\ f^{-1}(x) & = 3x-8\end{aligned}

**Example 1: Composite Functions**

Let f(x)=x-3 and g(x)=x^2

**[4 marks]**

Find:

a) fg(10) – we must find g(10) then apply f(x) to the answer.

g(10) = 10^2 = 100 so fg(10) = f(100) = 100 - 3 = 97.

b) gf(-4) – we must find f(-4) then apply g(x) to the answer.

f(-4) = -4-3 = -7 so gf(-4) = g(-7) = (-7)^2 = 49

c) an expression for fg(x) – we need to input g(x) into f(x). So, we get

fg(x) = f\left(g(x)\right) = g(x) - 3 = x^2 - 3

**Example 2: Inverse Functions**

Given that f(x) = 3x - 9, find f^{-1}(x)

**[3 marks]**

**Step 1:** Write the equation in the form x = f(y)

f(x) = 3x- 9 becomes x = 3y-9

**Step 2:** Rearrange to make y the subject

\begin{aligned}x &= 3y-9 \\ x+9 &= 3y \\ \dfrac{x+9}{3} &=y\end{aligned}

**Step 3:** Replace ywith f^{-1}(x)

\begin{aligned} \dfrac{x+9}{3} & = y \\ f^{-1}(x) & = \dfrac{x+9}{3}\end{aligned}

### Take an Online Exam

#### Functions Online Exam

#### Functions (Composite and Inverse) Online Exam

### Example Questions

**Question 1:** Let f(x) = \dfrac{10}{3x-5}

a) Find f(10)

b) Find f(2)

c) Find f(-1)

**[4 marks]**

a) Substituting x=10 into f(x), we find,

f(10) = \dfrac{10}{3(10)-5} = \dfrac{10}{25}= \dfrac{2}{5}=0.4

b) Substituting x=2 into f(x), we find,

f(10) = \dfrac{10}{3(2)-5} = \dfrac{10}{1}= 10

c) Substituting x=-1 into f(x), we find,

f(10) = \dfrac{10}{3(-1)-5} = \dfrac{10}{-8}=-\dfrac{5}{4} =1.25

**Question 2:** Let f(x) = \dfrac{15}{x} and g(x) = 2x - 5

a) Find fg(4)

b) Find gf(-30)

c) Find gf(x)

**[4 marks]**

a) Substituting x=4 into g(x), then substituting the result into f(x),

g(4) = (2\times 4) - 5 = 8 - 5 = 3

fg(4) = f(3) = \dfrac{15}{3} = 5

b) For gf(-30) we must first find f(-30) and then substitute the result into g(x),

f(-30) = \dfrac{15}{-30} = -\dfrac{1}{2}

gf(-30) = g(-\dfrac{1}{2}) = 2(-\dfrac{1}{2}) - 5 = -1 - 5 = -6

c) To find an expression for gf(x), substitute f(x) in for every instance of x in g(x),

gf(x) = 2(f(x)) - 5 = 2\times(\dfrac{15}{x}) - 5 = \dfrac{30}{x} - 5

**Question 3:** Find the inverse function of f(x) = \dfrac{5}{x-4}

**[3 marks]**

So, we need to write the function as y=\frac{5}{x-4} and rearrange this equation to make x the subject. Then, we will swap every y with an x – and vice versa.

We won’t be able to get x on its own whilst it’s in the denominator, so our first step will be multiplying both sides by (x-4):

y(x-4)=5

Then, divide both sides by y:

x-4=\dfrac{5}{y}

Finally, add 4 to both sides to make x the subject:

x=\dfrac{5}{y}+4

Now, swap each x with a y and vice versa to get

f^{-1}(x)=\dfrac{5}{x}+4

**Question 4:** Find the inverse function of g(x) = \dfrac{4}{x}+3

**[3 marks]**

So, we need to write the function as g=\frac{4}{x}+3 and rearrange this equation to make x the subject. Then, we will swap every g with an x – and vice versa.

The first step is to subtract 3 from both sides,

g-3=\dfrac{4}{x}+\cancel{3}-\cancel{3}

Then, multiply both sides by x:

x(g-3)=4

Finally, divide both sides by (g-3) to make x the subject:

x=\dfrac{4}{g-3}

Now, simply swap each x with a g and vice versa to get,

g^{-1}(x)=\dfrac{4}{x-3}

### Worksheets and Exam Questions

#### (NEW) Functions (The basics) Exam Style Questions - MME

Level 4-5 New Official MME#### (NEW) Functions (Composite and inverse) Exam Style Questions - MME

Level 6-8 New Official MME### Drill Questions

#### Functions - Drill Questions

### Learning resources you may be interested in

We have a range of learning resources to compliment our website content perfectly. Check them out below.