Functions Toolkit (AQA GCSE Further 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 $x$ in would give $2x+1$

• 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) = … 
  

  • 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

  • $\mathrm{f}\left(x\right)=3x–4$   

• In such cases, $x$ would be the input and $\mathrm{f}\left(x\right)$ would be the output

• Sometimes functions don’t have names like f and are just written as y = …

  • eg. $y=3x–4$

How does a function work?

• A function has an input $\left(x\right)$ and output $\left(\mathrm{f}\left(x\right)\mathrm{or}y\right)$

• Whatever goes in the bracket (instead of $x$)with f, replaces the $x$ on the other side

  • This is the input

• If the input is known, the output can be calculated

  • For example, given the function $\mathrm{f}\left(x\right)=2x+1$

    • $\mathrm{f}\left(3\right)=2\times 3+1=7$

    • $\mathrm{f}(-4)=2\times (-4)+1=-7$

    • $\mathrm{f}\left(a\right)=2a+1$

• If the output is known, an equation can be formed and solved to find the input

  • For example, given the function $\mathrm{f}\left(x\right)=2x+1$
    

    • If $\mathrm{f}\left(x\right)=15$, the equation $2x+1=15$ can be formed

    • Solving this equation gives an input of 7

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

  • you can use inequality signs if needed

  • you can exclude parts by saying "except" if needed

• For f(x) = 2x + 3 

  • the domain "x > 0" means only positive values of x can be inputted 

  • the domain "2 < x < 5 except 4" means only values of x between 2 and 5, except 4, can be inputted

    • this includes non-integers, like x = π

  • the domain "all real values" means any x can be inputted

• For $\mathrm{f}\left(x\right)=\sqrt{x-2}$ you cannot square root a negative number

  • the domain is x ≥ 2 

    • this is from solving x - 2 ≥ 0

• For $\mathrm{f}\left(x\right)=\frac{1}{x-5}$ you cannot divide by zero

  • the domain is all real values of x except 5

    • this is from solving x - 5 = 0

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

• Ranges are based on domains

• For 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 of y = 3x + 2 in the region x > 0

How do I solve problems involving the domain and range?

• You need to be able to deduce the range of a function from its expression and domain

• To find the range of g(x) = 3x² with the domain x ≥ 0...
  

  • ...sketch the graph for x ≥ 0 only (use a table-of-values if required)...

  

  • ...read-off the range by seeing which values of y are possible

    • Possible y values are y ≥ 0

    • rewrite "y" as "f(x)" when giving ranges

    • the range is f(x) ≥ 0

Piecewise functions

What is a piecewise function?

• A piecewise function is a single function with different parts across different domains

• The function $\mathrm{f}\left(x\right)=\{\begin{array}{l}{x}^2\\ 4\\ 9-x\end{array}\begin{array}{c}-2\le x\le 2\\ 2<x\le 5\\ 5<x\le 9\end{array}$ has three domains

  • the input 0.5 lies in $-2\le x\le 2$, so $\mathrm{f}\left(0.5\right)=0.5^2=0.25$

  • the input 3 lies in $2<x\le 5$ so $\mathrm{f}\left(3\right)=4$

  • the input 8.5 lies in $5<x\le 9$, so $\mathrm{f}\left(8.5\right)=9-8.5=0.5$

How do I sketch a piecewise function?

• Think of the shape of each part

  • f(x) = mx + c is a straight line, f(x) = k is horizontal at height k, f(x) = x² is quadratic etc

• Plot the coordinates of the end-points to help

  • for the domain a ≤ x ≤ b find the heights of the graph, f(a) and f(b)

• Not all parts have to "join up"

  • there may be a jump (discontinuity)

• For $\mathrm{f}\left(x\right)=\{\begin{array}{ll}x& 0\le x\le 1\\ 1& 1.5\le x\le 2\\ 0& \mathrm{otherwise}\end{array}$

  • it's y = x from x = 0 to 1, then horizontal at y = 1 from x = 1.5 to 2, then 0 everywhere else
    

    

  • Sketching helps see the range (all possible outputs on the y-axis), for this example it's $0\le \mathrm{f}\left(x\right)\le 1$

• A table of values can also be used

How do I find the equation of a piecewise function from a sketch?

• Build an equation in the form $\mathrm{f}\left(x\right)=\{\begin{array}{l}...\\ ...\\ ...\end{array}\begin{array}{c}a<x\le b\\ b<x\le c\\ c<x\le d\end{array}$

  • don't include a domain end-point twice (use ≤ with one part and < with another, in either order)

• Horizontal sections have the form f(x) = k, where k is the height

• Straight line sections can be thought of as a coordinate geometry problem

  • how do I find the equation of a straight line from (x₁, y₁) to (x₂, y₂)?

• Quadratic sections have two different equation forms

  • f(x) = (x - a)² + b is a positive quadratic (U-shape) with vertex (turning point) at (a, b)

    • the vertical line x = a is the line of symmetry

  • f(x) = (x - a)(x - b) is a positive quadratic (U-shape) with x-intercepts x = a and x = b