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Here we will learn about** function notation**, including different forms of function notation, how to evaluate functions for given values and how to manipulate algebraic expressions using functions.

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

**Function notation** is a way of expressing a relationship between two variables.

We are used to writing equations of straight lines in the form y = mx + c .

Using function notation we can write this as f(x) = mx + c , by replacing y with f(x) .

We can read this as “ the function f of x ”.

“f” can be thought of as the “name” of the function. We do not always have to use f for the name of the function; other common names include function g or function h.

The x is the input value known as the **independent variable**.

f(x) is the output value ( y -value) known as the **dependent variable**.

To use function notation we just substitute the values of x into the expression and evaluate it.

The diagram shows the value of a function for different x -values:

There are different types of function notation. As well as examples like g(x) = x^2 + 2 , we may also see it given as g : x → x^2 + 2 . This type of function notation that is more common in A level mathematics.

Function notation is also used in the table function of a scientific calculator. The table function is useful for finding values when graphing linear equations, quadratics, cubics and other polynomials.

In order to evaluate a function using function notation:

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

Get your free function notation worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEGet your free function notation worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE**Function notation** is part of our series of lessons to support revision on **functions in algebra**. You may find it helpful to start with the main functions in algebra lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

Find f(3) when f(x)= 4x - 1

**Write out the function for x using function notation, replacing the x with an empty set of brackets.**

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.**

Find g(-2) when g(x) = x^2 + 4

**Write out the function for x using function notation, replacing the x with an empty set of brackets.**

g\left( {} \right)={{\left( {} \right)}^{2}}+4

g\left( -2 \right)={{\left( -2 \right)}^{2}}+4

**Apply the correct operations to the number or term as appropriate and simplify.**

g\left( -2 \right)=8

Find h(8) when h(x) = x^3 - 3x

**Write out the function for ** x ** using function notation, replacing the x with an empty set of brackets.**

h\left( {} \right)={{\left( {} \right)}^{3}}-3\left( {} \right)

h\left( 8 \right)={{\left( 8 \right)}^{3}}-3\left( 8 \right)

**Apply the correct operations to the number or term as appropriate and simplify.**

h\left( 8 \right)=488

Find f(2m) when f(x) = 5x + 7

**Write out the function for x using function notation, replacing the** ** x with an empty set of brackets.**

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

f\left( 2m \right)=5\left( 2m \right)+7

**Apply the correct operations to the number or term as appropriate and simplify.**

f\left( 2m \right)=10m+7

Find g(a+3) when g(x) = x^2 - 1

**Write out the function for x using function notation, replacing the** ** x with an empty set of brackets.**

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

g\left( a+3 \right)={{\left( a+3 \right)}^{2}}-1

**Apply the correct operations to the number or term as appropriate and simplify.**

\begin{aligned}
& g\left( a+3 \right)={{a}^{2}}+6a+9-1 \\\\
& g\left( a+3 \right)={{a}^{2}}+6a+8
\end{aligned}

**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 mistake may be to think that f(2) means 2(x+3) .

Whereas the correct solution is:

\begin{aligned} &f(x)=x+3 \\ &f(2)=2+3 \\ &f(2)=5 \end{aligned}1. Find f(4) when f(x)=2x+8

12

16

4(2x+8)

8x+32

Replace the x with 4 and simplify.

2. Find g(7) when g(x)=5(x-1)

34

35(x-1)

5(x-7)

30

Replace the x with 7 and simplify.

3. Find h(-5) when h(x)=x^2+x+3

-27

33

23

-5x^2-5x-15

Replace the x with -5 and simplify.

4. Find f(3n) when f(x)=6(x+2)

18n+12

30n

30

18n+2

Replace the x with 3n , expand the brackets and simplify.

5. Find g(4k+1) when g(x)=5x-2

20k-2

23

20k+3

27

Replace the x with 4k+1 , expand the brackets and simplify.

6. Find h(a-2) when h(x)=x^2+5

a^2-4a+1

a^2-4a+9

a^2+4a+9

a^2+4a+1

Replace the x with a-2 , expand the brackets and simplify.

1. Given that f(x)=5x-2

(a) Find f(-4)

(b) Find x when f(x)=8

**(3 marks)**

Show answer

(a)

**(1)**

(b)

Equation formed 5x-2=8

**(1)**

Answer x=2

**(1)**

2. (a) If f(x)=4x+1 , write a simplified expression for f(2a-1)

(b) If h(x)=x^2+2x-3 , write a simplified expression for h(2m+3)

**(5 marks)**

Show answer

(a)

Substitution seen 4(2a-1)+1

**(1)**

8a-3

**(1)**

(b)

Substitution seen (2m+3)^2+2(2m+3)-3

**(1)**

Expanded brackets 4m^2+12m+9+4m+6-3

**(1)**

Simplified expression 4m^2+16m+12

**(1)**

3. Given that f(x)=x^2+2 and g(x)=3(x+4).

Find the value of x which satisfies f(x)=g(x)

**(4 Marks)**

Show answer

Set equal x^2+2=3(x+4)

**(1)**

Form quadratic x^2-3x-10=0

**(1)**

Factorise (x-5)(x+2)=0

**(1)**

Both solutions x=5,-2

**(1)**

You have now learned how to:

- Where appropriate, interpret simple expressions as functions with inputs and outputs

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