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Functions Toolkit (Edexcel IGCSE Maths)
Revision Note
Introduction to Functions
What is a function?
- A function is a combination of one or more mathematical operations that takes a set of numbers and changes them into another set of numbers
- It may be thought of as a mathematical “machine”
- For example, if the function (rule) is “double the number and add 1”, the two mathematical operations are "multiply by 2 (×2)" and "add 1 (+1)"
- Putting 3 in to the function would give 2 × 3 + 1 = 7
- Putting -4 in would give 2 × (-4) + 1 = -7
- Putting in would give
- The number being put into the function is often called the input
- The number coming out of the function is often called the output
What does a function look like?
- A function f can be written as f(x) = … or f : x ↦ …
- These two different types of notation mean exactly the same thing
- Other letters can be used. g, h and j are common but any letter can technically be used
- Normally, a new letter will be used to define a new function in a question
- For example, the function with the rule “triple the number and subtract 4” would be written
- or
- In such cases, would be the input and would be the output
- Sometimes functions don’t have names like f and are just written as y = …
- eg.
How does a function work?
- A function has an input and output
- Whatever goes in the bracket (instead of )with f, replaces the on the other side
- This is the input
- If the input is known, the output can be calculated
- For example, given the function
- For example, given the function
- If the output is known, an equation can be formed and solved to find the input
- For example, given the function
- If , the equation can be formed
- Solving this equation gives an input of 7
- For example, given the function
Worked example
A function is defined as .
(a)
Find .
The input is , so substitute 7 into the expression everywhere you see an .
Calculate.
(b)
Find .
The input is so substitute into the expression everywhere you see an .
Expand the brackets and simplify.
A second function is defined .
(c)
Find the value of for which .
Form an equation by setting the function equal to -16.
Solve the equation by first adding 4 to both sides, then dividing by 3.
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Domain & Range
How are functions related to graphs?
- Functions can be represented as graphs on x and y axes
- The x-axis values are the inputs
- The y-axis values are the outputs
- To see what graph to plot, replace f(x) = with y =
What is the domain of a function?
- The domain of a function is the set of all inputs that the function is allowed to take
- Domains can be described in words
- they must refer to x
- not y or f(x)
- you can use "not equal to" ≠ if needed
- you can use inequality signs if needed
- they must refer to x
- Examples of domains are below:
- f(x) = 3x + 2 takes any x value
- the domain is "all values of x"
- f(x) = takes any x value except 0 (you cannot divide by 0)
- the domain is "all values of x except 0", or simply "x ≠ 0"
- f(x) = takes any x value that are not negative (you cannot square root a negative)
- the domain is "x ≥ 0"
- f(x) = x2 takes any x value (negative x values are fine as inputs)
- the domain is "all values of x"
- f(x) = 3x + 2 takes any x value
- Some domains are restricted by choice
- f(x) = 3x + 2 with the domain 0 < x < 5
- This question wants to concentrate on that domain only (even though bigger domains exist)
- f(x) = 3x + 2 with the domain 0 < x < 5
- Some domains must exclude certain values (or sets of values)
- f(x) = must exclude x = 1 and x = -7 from any domain
- These two inputs make the function undefined (dividing by zero)
- f(x) = must exclude x < 3 from any domain
- Any input in x < 3 leads to square-rooting a negative
- f(x) = must exclude x = 1 and x = -7 from any domain
What is the range of a function?
- The range of a function is the set of all outputs that the function gives out
- Ranges can be described in words
- they must refer to f(x)
- not x or y
- you can use "not equal to" ≠ if needed
- you can use inequality signs if needed
- they must refer to f(x)
- Ranges are influenced by domains
- Examples of ranges are below:
- f(x) = 3x + 2 with domain x > 0
- The range is "f(x) > 2"
- This is because if the inputs are all greater than 0, the outputs will all be greater than 2
- This could be seen from a sketch or by substituting inputs of x > 0 into f(x)
- f(x) = x2 with domain "all values of x"
- The range is f(x) ≥ 0
- This is because all values of x get squared (so no negative outputs are created)
- f(x) = 3x + 2 with domain x > 0
How do I solve problems involving the domain and range?
- You need to be able to identify and explain any exclusions in the domain of a function
- You need to be able to deduce the range of a function from its expression and domain
- You may also be asked to sketch a graph of a function
- This could involve sketching parts of familiar graphs that are restricted because of the domain and exclusions
Examiner Tip
- A graph of the function can help “see” both the domain and range of function (a sketch can help if you have not been given a diagram)
Worked example
Two functions are given by
(a)
If the domain of function f is , find the range.
The domain is the set of inputs
Substitute x = 2 into f(x) to find its output
Substitute x = 2 into f(x) to find its output
Substitute x = 4 into f(x) to find its output
Think of f(x) = 10 - x as a graph
the graph of
This straight-line graph has a negative gradient
Between x = 2 and x = 4 the graph decreases from a height of 8 to a height of 6
Relate this to outputs
Between x = 2 and x = 4 the graph decreases from a height of 8 to a height of 6
Relate this to outputs
all outputs are between 6 and 8
Write down the range using f(x)
Remember that the inequality is "equal to" at x = 4, f(x) = 6
Remember that the inequality is "equal to" at x = 4, f(x) = 6
the range is
(b)
Write down the value of x that must be excluded from the domain of function g.
An input cannot cause the function to divide by zero
Find out when "dividing by zero" would happen
Find out when "dividing by zero" would happen
Solve to find this value of x (the one that must be excluded)
must be excluded
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