Relations vs. Functions

A relation is any set of ordered pairs $(x,y)$ — a rule, a table, a graph, or a list of points that connects inputs to outputs, with no restriction at all. A function is a relation with one extra condition: every input value ($x$) is paired with exactly one output value ($y$). It's fine for two different inputs to share the same output (many students can get the same test mark), but it is not fine for one input to produce two different outputs. Why insist on this? Because a function is meant to be predictable: if you know the input, you know the output with no ambiguity. A student number that could belong to two different students would be useless for looking anyone up. Functions where no two inputs share an output (like $y=2x+1$) are called one-to-one; functions where several inputs share an output (like $y=x^2$, where $-3$ and $3$ both give $9$) are many-to-one. Both kinds are functions. Any relation in which some input has several outputs — one-to-many or many-to-many, like a circle — is not a function.

On a graph, "one output per input" means: pick any vertical line $x=k$ — it should cross the graph at most once. If some vertical line crosses the graph two or more times, then that $x$-value has two different $y$-values, so the graph is not a function. This is the vertical line test, and it's the fastest way to check a graph. It works because every point on a vertical line shares the same $x$-coordinate: two crossing points means two ordered pairs with the same input and different outputs. You only need one failing line to reject a graph, but to accept a graph you must be convinced that every vertical line crosses at most once. "At most once" includes zero times — a vertical line that misses the graph just means that $x$-value is not in the domain.

• A circle ($x^2+y^2=r^2$) fails the vertical line test — it is not a function.
• A sideways parabola $x=y^2$ fails the vertical line test — it is not a function.
• A straight (non-vertical) line always passes — it is a function.
• A vertical line such as $x=4$ is the worst case: every point has the same input, so it is not a function.
• $y=x^2$ passes — every $x$ gives one $y$, even though two different $x$-values can share a $y$ (e.g. $(-2,4)$ and $(2,4)$).

MCR3U (1.1) - Relations vs Functions - Grade 11 Functions

Instead of writing $y=2x-3$, we often write $f(x)=2x-3$. This is function notation: $f(x)$ is read "$f$ of $x$" and it means "the output of function $f$ when the input is $x$." It is a name for the output — it does not mean $f$ multiplied by $x$. Different letters ($f$, $g$, $h$, ...) let you talk about several functions in the same problem without confusion, and the notation packs the input and output into one symbol: $f(3)=7$ says "the input $3$ gives the output $7$," which is the point $(3,7)$ on the graph. To evaluate $f(a)$ for some value or expression $a$, substitute $a$ for every occurrence of $x$ in the rule, then simplify. This works whether $a$ is a number, another variable, or a whole expression like $a+1$ — put the substituted value in brackets so that signs and exponents apply to all of it.

1.2 Function Notation f(x)=

Function notation earns its keep in real contexts. If a streaming service charges $C(n)=15+2.5n$ dollars for a month with $n$ movie rentals, then $C(8)=15+2.5(8)=35$ reads "renting 8 movies costs 35 dollars." Two different questions come up constantly, and they run in opposite directions. Evaluating ("find $C(8)$") gives you the input and asks for the output — substitute. Solving ("for what $n$ is $C(n)=40$?") gives you the output and asks for the input — set the rule equal to the output and solve the equation: $15+2.5n=40$, so $n=10$. For a many-to-one function, solving can produce more than one input; for $f(x)=x^2$, the equation $f(x)=9$ has two solutions, $x=3$ and $x=-3$. That is not a contradiction — two inputs sharing one output is allowed.