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Function Basics (College Board AP® Precalculus): Study Guide

Roger B

Written by: Roger B

Reviewed by: Mark Curtis

Updated on

Function basics

What is a function?

  • A function is a mathematical relation that maps a set of input values to a set of output values

    • Each input value is mapped to exactly one output value

    • E.g. in the relation space f open parentheses x close parentheses equals x squared plus 1, space f is a function

      • because for any value of x you put in, you get exactly one output value out

  • If a single input value can produce more than one output value, then the relation is not a function

    • E.g. the relation x squared plus y squared equals 25 is not a function of x

      • because for the input x equals 3

      • there are two output values: y equals 4 and y equals negative 4

What are the domain and range of a function?

  • The domain of a function is the set of all input values

    • The variable representing input values is called the independent variable

  • The range of a function is the set of all output values

    • The variable representing output values is called the dependent variable

      • It is called the dependent variable because its value depends on the value chosen for the independent variable

  • E.g. consider the function space f given by space f open parentheses x close parentheses equals x squared, defined for negative 3 less or equal than x less or equal than 3

    • The domain is negative 3 less or equal than x less or equal than 3

    • The range is 0 less or equal than space f open parentheses x close parentheses less or equal than 9

      • The square of any real number is positive, so space f open parentheses x close parentheses equals x squared cannot take on values less than zero

    • x is the independent variable

      • and space f open parentheses x close parentheses (or space y, where space y equals f open parentheses x close parentheses) is the dependent variable

    • The function space f takes any value x in its domain

      • and maps it to the value space f open parentheses x close parentheses in its range

Diagram showing function f mapping element x from Domain to f(x) in Range, represented by two circles connected by an arrow.
Domain and range of a function f

How can a function be represented?

  • The input and output values of a function vary in tandem according to the function rule

  • This rule can be expressed in four ways:

    • Analytically (i.e. as an equation), e.g. space f open parentheses x close parentheses equals 3 x minus 5

    • Graphically, as a curve or line on a set of axes

    • Numerically, as a table of input-output pairs

    • Verbally, as a description in words, e.g. "the output is five less than three times the input"

  • These are all different ways of representing the same function

    • Being able to move between representations is a key skill in AP® Precalculus

How can you tell if a graph represents a function?

  • Use the vertical line test

    • If any vertical line drawn on the graph crosses the curve at more than one point, then the graph does not represent a function

    • This is because a vertical line represents a single input value, and crossing more than once means that input is mapped to more than one output

Two graphs: left shows a curve passing the vertical test as a function; right shows a circle failing the test, so not a function.
The vertical line test for a function

Worked Example

A store is monitoring the temperature, in degrees Fahrenheit, inside a refrigerator unit after a power outage. The following table records the temperature T, in degrees Fahrenheit, at time t, in hours after the outage.

bold italic t (hours)

0

1

2

3

4

T open parentheses t close parentheses (°F)

35

38

44

53

65

(a) Find T open parentheses 3 close parentheses. Describe the meaning of this value in the context of the problem.

Answer:

T open parentheses 3 close parentheses equals 53. This means that 3 hours after the power outage, the temperature inside the refrigerator unit is 53°F.

(b) Describe how the temperature and time are varying in tandem. Is the temperature increasing or decreasing as time increases?

Answer:

As the input values (time) increase, the output values (temperature) also increase. The temperature and time are increasing in tandem, i.e. as more time passes since the outage, the temperature inside the refrigerator gets higher.

(c) In the context of the problem, identify the independent variable and the dependent variable.

Answer:

The independent variable is t (time in hours), and the dependent variable is T (temperature in °F).

Worked Example

The function space f is defined by space f open parentheses x close parentheses equals x squared minus 4 x plus 7, with domain open curly brackets 1 comma space 2 comma space 3 comma space 4 comma space 5 close curly brackets.

Find the range of space f.

Answer:

The domain of this function is only the five values given.

Find the output value for each input value in the domain:

space f open parentheses 1 close parentheses equals open parentheses 1 close parentheses squared minus 4 open parentheses 1 close parentheses plus 7 equals 1 minus 4 plus 7 equals 4

space f open parentheses 2 close parentheses equals open parentheses 2 close parentheses squared minus 4 open parentheses 2 close parentheses plus 7 equals 4 minus 8 plus 7 equals 3

space f open parentheses 3 close parentheses equals open parentheses 3 close parentheses squared minus 4 open parentheses 3 close parentheses plus 7 equals 9 minus 12 plus 7 equals 4

space f open parentheses 4 close parentheses equals open parentheses 4 close parentheses squared minus 4 open parentheses 4 close parentheses plus 7 equals 16 minus 16 plus 7 equals 7

space f open parentheses 5 close parentheses equals open parentheses 5 close parentheses squared minus 4 open parentheses 5 close parentheses plus 7 equals 25 minus 20 plus 7 equals 12

The range is the set of output values

The range is open curly brackets 3 comma space 4 comma space 7 comma space 12 close curly brackets

Examiner Tips and Tricks

In the second Worked Example, note that although space f open parentheses 1 close parentheses equals f open parentheses 3 close parentheses equals 4, the value 4 is only listed once in the range.

Two different inputs mapping to the same output is allowed for a function.

  • It is only one input mapping to two different outputs that is not allowed

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Author
Reviewer
Roger B

Author: Roger B

Expertise: Maths Content Creator

Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

Mark Curtis

Reviewer: Mark Curtis

Expertise: Maths Content Creator

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.