Functions and Graphs
2.2 Functions
Relations Versus Functions
A relation can pair inputs and outputs in many ways. A function is stricter: each input gets exactly one output.
Definition: Relation and Function
A relation is any set of ordered pairs.
A function is a relation where each input $x$ is paired with exactly one output $y$.
Fact: Vertical Line Test
A graph represents a function if and only if every vertical line intersects the graph at most once.
Example: Decide if a mapping is a function
Consider the pairs
$${(-1,4),(0,1),(2,1),(2,5)}. $$
Is this a function?
Show Solution
No. The input $x=2$ is paired with two outputs, $1$ and $5$.
That violates the function rule.
MyOpenMath: Practice: function or not a function (placeholder)
Function Notation and Evaluation
Function notation helps us read expressions as outputs of a rule.
Definition: Function Notation
If a function is named $f$, then $f(x)$ means the output at input $x$.
Example: Evaluate a polynomial function
Let
$$f(x)=x^2-3x+1.$$
Find $f(4)$ and $f(-2)$.
Show Solution
Substitute directly:
$$f(4)=4^2-3(4)+1=16-12+1=5,$$ $$f(-2)=(-2)^2-3(-2)+1=4+6+1=11.$$
Example: Evaluate a piecewise function
Let
$$ f(x)= \begin{cases} 2x+1, & x<0,\ x^2, & x\ge 0. \end{cases} $$
Find $f(-3)$ and $f(2)$.
Show Solution
Since $-3<0$, use $2x+1$:
$$f(-3)=2(-3)+1=-5.$$
Since $2\ge0$, use $x^2$:
$$f(2)=2^2=4.$$