GCSEFurther MathsAQARevision Notes2. Algebra & FunctionsFunctionsFunctions Toolkit

Functions Toolkit (AQA GCSE Further Maths)

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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 2 x space plus space 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
    • straight f left parenthesis x right parenthesis space equals space 3 x space – space 4   
  • In such cases, x would be the input and straight f open parentheses x close parentheses would be the output
  • Sometimes functions don’t have names like f and are just written as y = …
    • eg. y space equals space 3 x space – space 4

How does a function work?

  • A function has an input open parentheses x close parentheses and output left parenthesis straight f open parentheses x close parentheses space or space space y right parenthesis
  • 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 straight f left parenthesis x right parenthesis space equals space 2 x space plus space 1
      • straight f left parenthesis 3 right parenthesis space equals space 2 space cross times space 3 space plus space 1 equals 7
      • straight f left parenthesis negative 4 right parenthesis space equals space 2 space cross times space left parenthesis negative 4 right parenthesis space plus space 1 space equals space minus 7
      • straight f left parenthesis a right parenthesis space equals space 2 a space plus space 1
  • If the output is known, an equation can be formed and solved to find the input
    • For example, given the function straight f left parenthesis x right parenthesis space equals space 2 x space plus space 1
      • If straight f left parenthesis x right parenthesis space equals space 15, the equation 2 x space plus space 1 space equals space 15 can be formed
      • Solving this equation gives an input of 7

Worked example

A function is defined as straight f open parentheses x close parentheses space equals space 3 x to the power of 2 space end exponent minus space 2 x space plus space 1.

(a)
Find straight f open parentheses 7 close parentheses.
  

The input is x space equals space 7, so substitute 7 into the expression everywhere you see an x.

straight f open parentheses 7 close parentheses space equals space 3 open parentheses 7 close parentheses squared space minus space 2 open parentheses 7 close parentheses space plus space 1  

Calculate.

table row cell straight f open parentheses 7 close parentheses space end cell equals cell space 3 open parentheses 49 close parentheses space minus space 14 space plus space 1 end cell row blank equals cell space 147 space minus space 14 space plus space 1 end cell end table  

bold f stretchy left parenthesis 7 stretchy right parenthesis bold space bold equals bold space bold 134

(b)

Find straight f open parentheses x space plus space 3 close parentheses.

 

The input is x space equals space x space plus space 3 so substitute space x space plus space 3 into the expression everywhere you see an x.

straight f open parentheses x space plus space 3 close parentheses space equals space 3 open parentheses x space plus space 3 close parentheses squared space minus space 2 open parentheses x space plus space 3 close parentheses space plus space 1 

   Expand the brackets and simplify.

table row cell straight f open parentheses x space plus space 3 close parentheses space end cell equals cell space 3 open parentheses x squared space plus space 6 x space plus space 9 close parentheses space minus space 2 open parentheses x space plus space 3 close parentheses space plus space 1 end cell row blank equals cell space 3 x squared space plus space 18 x space plus space 27 space minus space 2 x space minus space 6 space plus space 1 end cell row blank equals cell space 3 x squared space plus space 16 x space plus space 22 end cell end table 

A second function is defined straight g open parentheses x close parentheses equals space 3 x space – space 4.

(c)

Find the value of x for which straight g open parentheses x close parentheses equals space minus 16.
  

Form an equation by setting the function equal to -16.

table row cell 3 x space minus space 4 space end cell equals cell space minus 16 end cell end table 

Solve the equation by first adding 4 to both sides, then dividing by 3. 

table row cell 3 x space minus space 4 space end cell equals cell space minus 16 end cell row cell 3 x space end cell equals cell space minus 12 end cell row cell x space end cell equals cell space minus 12 over 3 end cell end table 

bold italic x bold space bold equals bold space bold minus bold 4

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 = ...

DRE Notes fig1, downloadable IGCSE & GCSE Maths revision notes

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 straight f open parentheses x close parentheses equals square root of x minus 2 end root you cannot square root a negative number
    • the domain is x ≥ 2 
      • this is from solving x - 2 ≥ 0
  • For straight f open parentheses x close parentheses equals fraction numerator 1 over denominator x minus 5 end fraction 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)...

    DRE Notes fig5, downloadable IGCSE & GCSE Maths revision notes

    • ...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

Exam Tip

  • A graph / sketch of the function helps to “see” the domain on the x-axis and range on the y-axis

Worked example

Two functions are given by
 

straight f open parentheses x close parentheses equals 3 x plus 5 space space space space space space space space space space space space space space straight g open parentheses x close parentheses equals 9 minus x

(a)
If the domain of function f is 2 less than x less or equal than 4, find the range.
 

Sketch the graph of straight f open parentheses x close parentheses by substituting straight f open parentheses x close parentheses for yand sketching the linear graph y space equals space 3 x space plus space 5.

The domain of the function is 2 space less than space x space less or equal than space 4 so only draw the graph between these points. 

Substitute x space equals space 2 and x space equals space 4 into the function to find the endpoints of the range. 

straight f open parentheses 2 close parentheses space equals space 3 open parentheses 2 close parentheses space plus space 5 space equals space 11 straight f open parentheses 4 close parentheses space equals space 3 open parentheses 4 close parentheses space plus space 5 space equals space 17

5sMHl9UL_aqa-fm-domain-and-range-rn-diagram-we-1
The range is the y values that the graph goes from and to.

Use the domain to decide whether the range has a strict inequality (≤ or ≥) or a non-strict inequality (< or >). 
The domain is greater than 2 but less than or equal to 4 so the range is greater than 11, but less than or equal to 17.

When writing the range you must use the straight f open parentheses x close parentheses notation in the final answer.

 
(b)
If the range of g is 4 less than straight g open parentheses x close parentheses less or equal than 6, find the domain.
 

Sketch the graph of straight g open parentheses x close parentheses by substituting straight g open parentheses x close parentheses for yand sketching the linear graph y space equals space 9 space minus space x.

The range of the function is 4 space less than space straight g open parentheses x close parentheses space less or equal than space 6 so substitute y space equals space 4 and y space equals space 6 into the function and solve to find the endpoints of the domain. 

For y space equals space 4,

table row cell 9 space minus space x space end cell equals cell space 4 end cell row cell negative x space end cell equals cell space minus 5 end cell row cell x space end cell equals cell space 5 end cell end table

For y space equals space 6,

table row cell 9 space minus space x space end cell equals cell space 6 end cell row cell negative x space end cell equals cell space minus 3 end cell row cell x space end cell equals cell space 3 end cell end table

aqa-fm-domain-and-range-rn-diagram-we-2
The domain is the x values that the graph goes from and to.

Use the given range to decide whether the range has a strict inequality (≤ or ≥) or a non-strict inequality (< or >). 
The range is greater than 4 but less than or equal to 6 so the domain is greater than or equal to 3, but less than 5. (Make sure you look at the coordinates to check which part of the domain goes with which part of the range).

Note that this would be difficult to see without sketching the graph, as the function decreases. 

bold 3 bold space bold less or equal than bold space bold italic x bold space bold less than bold space bold 5

 

Piecewise Functions

What is a piecewise function?

  • A piecewise function is a single function with different parts across different domains
  • The function straight f open parentheses x close parentheses equals open curly brackets table row cell x squared end cell row 4 row cell 9 minus x end cell end table close space table row cell negative 2 less or equal than x less or equal than 2 end cell row cell 2 less than x less or equal than 5 end cell row cell 5 less than x less or equal than 9 end cell end table has three domains
    • the input 3 lies in 2 less than x less or equal than 5 so straight f open parentheses 3 close parentheses equals 4
    • ff open parentheses 8.5 close parentheses uses the third part to become straight f open parentheses 0.5 close parentheses then uses the first part to become 0.25

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 straight f left parenthesis x right parenthesis equals stretchy left curly bracket table row x cell 0 less or equal than x less or equal than 1 end cell row 1 cell 1.5 less or equal than x less or equal than 2 end cell row 0 otherwise end table
    • it's y = x from x = 0 to 1, then horizontal at y = 1 from x = 1.5 to 2, then 0 everywhere else
      1-3-1-ial-fig1-pdf-graph
    • Sketching helps see the range (all possible outputs on the y-axis), for this example it's 0 less or equal than straight f open parentheses x close parentheses less or equal than 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 straight f open parentheses x close parentheses equals open curly brackets table row cell... end cell row cell... end cell row cell... end cell end table close space space table row cell a less than x less or equal than b end cell row cell b less than x less or equal than c end cell row cell c less than x less or equal than d end cell end table
    • 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

Worked example

Sketch the function straight f open parentheses x close parentheses equals open curly brackets table row cell x plus 3 end cell row cell open parentheses x minus 1 close parentheses open parentheses x minus 3 close parentheses end cell row 3 end table close space space table row cell negative 3 less than x less or equal than 0 end cell row cell 0 less than x less or equal than 4 end cell row cell 4 less than x less or equal than 6 end cell end table

Consider each part of the function separately.

From x space equals negative 3 to x space equals space 0 the function is a linear graph, with a gradient of 1 and a y-intercept of 3. Find the endpoints of this part of the function.

At x space equals negative 3,  y space equals negative 3 space plus space 3 space equals space 0
At x space equals space 0,   y space equals 0 space plus space 3 space equals space 3

Plot the points (-3, 0) and (0, 3) and then draw a straight line between them. Check that its gradient is one. 

From x space equals space 0 to x space equals space 4 the function is a positive quadratic graph (u shape).

Find the endpoints of the quadratic graph by substituting x space equals space 0 and x space equals space 4 into the function and finding the corresponding y values.

At  x space equals space 0,  y space equals space open parentheses 0 minus 1 close parentheses open parentheses 0 minus 3 close parentheses space equals space 3
At x space equals space 4,   y space equals space open parentheses 4 minus 1 close parentheses open parentheses 4 minus 3 close parentheses space equals space 3

Find the x-intercepts of the quadratic graph by setting each bracket to 0 and solving.

Let open parentheses x minus 1 close parentheses space equals space 0,  x space equals space 1
Let open parentheses x space minus space 3 close parentheses space equals space 0,   x space equals space 3

Plot the points (0, 3), (1, 0) (3, 0) and (4, 3) and draw a smooth u-shaped curve through them.

The final part of the function is a horizontal line from the coordinates (4, 3) to (6, 3).

ygQMUXOO_aqa-fm-piecewise-functions-rn-diagram-we-1

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    Mark

    Author: Mark

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.