GCSE Maths Algebra

Functions

# Functions in Algebra

Here we will learn about functions in algebra, including function machines, composite functions and inverse functions.

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

## What are functionsin algebra?

Functions in algebra are used to describe the operation being applied to an input in order to get an output.

### Function machines

Functions can be presented using a function machine.

Step-by-step guide: Function machines

### Function notation

Functions can be described using function notation.

The f in f(x) is known as the function that being applied to a variable x . Other letters such as g and h are also commonly used.

E.g.

Let’s look at a function f described by a function machine:

Then let’s rewrite this using function notation.

Step-by-step guide: Function notation

### Composite functions

We can combine functions to make composite functions.

Step-by-step guide: Composite functions

### Inverse functions

We can also find inverse functions.

Step-by-step guide: Inverse functions

## How to use function machines

In order to solve an equation using a function machine:

1. Consider the order of operations being applied to the unknown.
2. Draw a function machine starting with the unknown as the input and the value of the equation as the output.
3. Work backwards, applying inverse operations to find the unknown input.

## Function machines example

### Example 1: Solving a two step equation using a function machine

Solve \frac{x}{3}+7=3

1. Consider the order of operations being applied to the unknown.

The order of operations is \div 3 , then +7 .

2Draw a function machine starting with the unknown as the input and the value of the equation as the output.

3Work backwards, applying inverse operations to find the unknown input.

x=\left( 3-7 \right)\times 3

x=-12

## How to use function notation

In order to evaluate a function using function notation:

1. Write out the function for x using function notation, replacing the x with an empty set of brackets (parentheses).
2. Replace the x in the function with the number or algebraic term in the brackets next to the name of the function.
3. Apply the correct operations to the number or term as appropriate and simplify.

## Function notation examples

### Example 2: Evaluating functions for numerical values

Find f(2) when f(x) = 3x + 2

f\left( {} \right)=3\left( {} \right)+2

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

f\left( 2 \right)=8

### Example 3: Evaluating functions for algebraic expressions

Find f(4m) when f(x) = 5x − 7

f\left( {} \right)=5\left( {} \right)-7

f\left( 4m \right)=5\left( 4m \right)-7

f\left( 4m \right)=20m-7

## How to evaluate composite functions

In order to evaluate composite functions:

1. 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.

## Composite functions example

### Example 4: Evaluating composite functions

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

g\left( 2 \right)={{\left( 2 \right)}^{2}}+1

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

f(5) = 15 , so fg(2) = 15 .

## How to find composite functions

In order to find composite functions:

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

## Composite function example

### Example 5: Finding composite functions

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

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

\begin{aligned} & fg\left( x \right)=2{{x}^{2}}+4-1 \\\\ & =2{{x}^{2}}+3 \end{aligned}

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

## How to find inverse functions

In order to find an inverse function.

1. Write out the expression for the original function using a y instead of the x . Set this expression equal to x .
2. Rearrange the equation to make y the subject.
3. Write your inverse function using the f^{-1} notation.

## Inverse function examples

### Example 6: Finding the inverse of a linear function

If f(x) = 3x + 7 , find f^{-1}(x) .

3y+7=x

\begin{aligned} & 3y+7=x \\\\ & 3y=x-7 \hspace{2cm} \text{Subtract } 7 \text{ from both sides of the equation} \\\\ & y=\frac{x-7}{3} \hspace{2cm} \text{Divide both sides by } 3 \end{aligned}

{{f}^{-1}}\left( x \right)=\frac{x-7}{3}

### Common misconceptions

• Not working backwards when using inverse functions

When using two-step function machines or others with more operations to solve equations, a common error is to forget to work backwards. The inverse operations are used but in the wrong order.

• Not using the correct order of operations when drawing a function machine

A common error is to not follow the correct order of operations when creating a function machine for an equation.

For example, for the equation 2x − 1 = 7 , the multiplication by two takes place before subtracting one.

• Function notation is mistaken for a product

It is common for f(x) to be thought of as “ f times x rather than “ f of x ”. This confusion can lead to incorrect evaluations of values.

E.g.

When finding f(2) when f(x)=x+3 , a student may think f(2) means 2(x+3) .

The correct solution is

\begin{aligned} &f(x)=x+3 \\\\ &f(2)=2+3 \\\\ &f(2)=5 \end{aligned}

• 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.

Students can apply them left to right by mistake. Thinking of fgh(x) as f[g(h(x))] helps. h is the innermost function and we need to work outwards.

• The inverse function notation means “ 1 over f(x)

A common error is to think the inverse function will be the reciprocal of the function because of the “power of -1 ” notation.

Eg if f(x)=3x , an incorrect expression for f^{-1} would be (3x)^{-1} or \frac{1}{3x}.

### Practice functions in algebra questions

1.  Find the missing Output and missing Input for the function machine.

a=12,\;b=25

a=21,\;b=16

a=21,\;b=2.5

a=4.5,\;b=40

Work forwards to find a and backwards, using inverse operations, to find b .

2. Select the correct function machine and solution for the equation:

2x+6=10

x is the input, the first operation is multiplying by 2 , the second operation is adding 6 , the output is 10.

3.  Find f(7) when f(x)=2x-8

7.5

6

7(2x-8)

14x-8

Replace the x with 7 and simplify.

4.  Find g(3k+2) when g(x)=6x-5

18k-5

18k-10

18k+12

18k+7

Replace the x with 3k+2, expand the brackets and simplify.

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

2x^2-2

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

4x^2-12x+10

4x^2-12x-8
fg(x)=f(2x+3)=(2x-3)^2+1.

6.  If g(x)=\frac{x}{2}-7 , find g^{-1}(x):

2x+14

2x-7

2x-14

2x+7

Rearranging the formula, first add 7 , then multiply by 2.

### Functions in algebra GCSE questions

1.

(a) Find the output if the input is 10.

(b) Write a simplified expression for the input if the output is 24n-32

(4 marks)

(a)

56

(1)

(b)

\frac{24n-32}{8}

(1)

3n-4+3

(1)

3n-1

(1)

2. If f(x)=5-2x and g(x)=2x+6

(a) Find x when f(x)=-9

(b) Write an expression for gf(x)

(5 marks)

(a)

Sight of 2x=14

(1)

x=7

(1)

(b)

2(5-2x)+6

(1)

10-4x+6

(1)

16-4x

(1)

3.  g(x)=\frac{x-3}{4} and h(x)=\frac{x}{2}+3

Solve when g^{-1}(x)=h^{-1}(x)

(4 marks)

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

(1)

h^{-1}(x)=2(x-3)   or  2x-6

(1)

Set equal 4x+3=2x-6

(1)

(1)

## Learning checklist

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
• interpret the reverse process as the ‘inverse function’

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