Working with Functions

Starting point: Years 9 and 10

Rules, tables and graphs

A rule like "double it and add 3" can be written as $y=2x+3$. A table of values lists inputs and outputs:

Plot the points and join them to see the graph. Any set of ordered pairs $(x,y)$ is called a relation.

What makes a function

A function is a relation where every $x$ gives exactly one $y$.

Function notation

Instead of $y=2x+3$, write $f(x)=2x+3$. Read it as "f of x". To find $f(4)$, put 4 in for $x$:

$$f(4)=2(4)+3=11$$

So $f(4)=11$ means the point $(4,11)$ is on the graph.

Quick summary

• A function gives exactly one output for each input. Use the vertical line test.
• $f(a)$ means substitute $a$ for $x$.
• Domain: allowed inputs. Range: possible outputs. Watch for dividing by zero and roots of negatives.
• $f(g(x))$: do $g$ first. Order matters.
• Even: $f(-x)=f(x)$, symmetric in the y-axis. Odd: $f(-x)=-f(x)$, symmetric about the origin.

Quiz