Function Basics (College Board AP® Precalculus): Revision Note

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 $f\left(x\right)=x^2+1$, $f$ 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 $x^2+y^2=25$ is not a function of $x$

    • because for the input $x=3$

    • there are two output values: $y=4$ and $y=-4$

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 $f$ given by $f\left(x\right)=x^2$, defined for $-3\le x\le 3$

  • The domain is $-3\le x\le 3$

  • The range is $0\le f\left(x\right)\le 9$

    • The square of any real number is positive, so $f\left(x\right)=x^2$ cannot take on values less than zero

  • $x$ is the independent variable

    • and $f\left(x\right)$ (or $y$, where $y=f\left(x\right)$) is the dependent variable

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

    • and maps it to the value $f\left(x\right)$ in its range

When are two functions equal?

• Two functions $f$ and $g$ are equal when both of the following are true

  • they have the same domain

  • for every input value $a$ in that domain, they produce the same output value

    • that is, $f\left(a\right)=g\left(a\right)$ for every $a$ in the domain

  • Both conditions are required

• Two functions with the same rule but different domains are not equal

  • E.g. $f\left(x\right)=x^2$ with domain all real numbers, and $g\left(x\right)=x^2$ with domain $0\le x\le 3$, are not equal functions, because their domains are different

• Two functions that are written differently can still be equal, provided they share the same domain and give the same output for every input in it

  • E.g. $f\left(x\right)=(x+1{)}^2$ and $g\left(x\right)=x^2+2x+1$, both with domain all real numbers, are equal functions

What are the image and preimage?

• These two terms describe precisely how a function pairs inputs with outputs

• The image of an input value is the single output value that the function rule produces for that input

  • E.g. for the function $f\left(x\right)=x^2$, the image of the input $3$ is the output $f\left(3\right)=9$

• The preimage of an output value is the set of all input values for which the function rule produces that output

  • E.g. for the function $f\left(x\right)=x^2$, the preimage of the output $9$ is the set $\{-3, 3\}$

    • because $f(-3)=9$ and $f\left(3\right)=9$

• Note the difference between the two terms

  • An input value always has exactly one image

    • because each input maps to exactly one output

  • An output value may have a preimage containing one, several, or no input values

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. $f\left(x\right)=3x-5$

  • 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