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

Functionbecause substituting any number into x will produce a unique y value.2. y = x²

**Function**

3. 6y + 2x = 6

**Function.** Isolating for y, you get:

null

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.

Answer: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).

Answers: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.