Introduction to Functions

A function is a mathematical relationship between two or more variables. Like an equation, a function contains one or more input variables that work together to produce exactly one output. However, unlike equations, they are given their own special notation called functional notation. For example, a relationship between the input x and its output would be represented as f (x) – read “f of x”. Arguably, the word is synonymous with equation but there’s a clear difference. For instance, x + 3 = 10 and z = 3 are both equations because they show an equivalence. These equations are not functions because there isn’t an output variable. Thus a function may be in the form of an equation, but not every equation is a function.

Let’s practice writing an equation in functional notation. Starting with y = 10 + x, this would be written as:

Set y = f(x)

∴   f(x) = 10 + x   – notice that y isn’t required.

To test whether an equation is a function, you must isolate y (if not already) and see if for every x input, only one output exists. Here are four examples:

Question: Discuss whether each equation below is also considered a function.

1.   y = 3x + 4

Function because substituting any number into x will produce a unique y value.

2.   y = x²




3.   6y + 2x = 6


Function. Isolating for y, you get:


This situation is similar to (1).


4.   y² + 5x = −7

Not a function. Isolating for y gets you: null

Substituting a value into x will produce 2 outputs, one that’s negative and one that’s positive.

Another thing that’s interesting about functions is that they all pass the vertical line test. The vertical line test states that if a vertical line comes in contact with two points of the plotted object, then the relation is not a function.

For instance, the graph on your left passes the test, while the one on the right fails.

Domain and Range

Nearly every function that models a real-life system has a domain and range that it cannot surpass. The domain refers to all possible input values, while the range refers to the outputs. For example, let’s say we were coding an application for a phone that displays the day of the week (Monday through Sunday) when the day (between 1 and 365) and year (between 1900 to current) are inputted. The domain for this function would be the possible inputs: 1 to 365 and 1900 to the current year. The range would be limited to Monday through Sunday.

The notion can be applied to functions written mathematically, as well. The limits affecting y (the output) is referred to as range, while the limits affecting x (input) is referred to as the domain. Think of the domain as all the values of x you’re allowed to plugin into a function without getting an error. Interestingly, had you chosen x to represent the days and z for the year, the functional notation would have to include both x and z like this f(x, z) = week day.

Let’s look a simple sample problem related to domain and range:

Question: State the domain and the range of the following relation. After finding the domain and range, plot the relation and state whether or not it is a function.


The domain consists of all x values between –4 and 1, represented as: –4 ≤ x ≤ 1.

The range consists of only –3.

Substituting into Functions

Substituting values into functions is no different than substituting a value into an equation containing unknowns.

Question: Substitute the given numerical value into each function:

a) If f(x) = 2x² + 4, find f(3).

b) If f(x, y) = 5x + 2y, find f(1, 2).


a) f(3) = 2( 3 )² + 4 = 22

b) f(1, 2) = 5( 1 ) + 2( 2 ) = 9

The video below shows a few more examples of varying difficulty similar to the ones above.

You’re also encouraged to watch part two, where multiple functions are evaluated and combined.