# Functions

In Mathematics, an equation is a statement that shows two things are equal. For example: 10 = 5 x 2, is an equation, as both sides of the equals sign are equal to 10.

Frequently, we replace exact numbers with variables, so you might find that 10 = 5 + x. We know that both sides of the equation must be equal & hence can determine that x = 5.

A function is a certain type of equation; it’s an equation that can only output a single value for Y given a single value for X (that is to say, the input must have a one-to-one relationship with the output). An example of a function is f(x) = 2x. If we define x as 4; then f(4) = 8.

A function has a function domain, that’s all the possible input values that can be ingested into the function. It also has a range; which is all the possible output values.

Below are some domain restrictions. For example:

• With the function in the bottom right, y must be a positive number, as we cannot square root a negative number. Hence, the domain will be anything >0.
• In the top left function, we have explicitly stated that x cannot equal zero. That’s because division by zero is an operation for which you cannot find an answer and is therefore disallowed.

### Is it a function?

So then, how do we prove something is a function, rather than an equation? Well, we have to prove there is a single Y output for an input X. We can do this algebraically, or using the straight line test.

Vertical Line Test

Below is an example of 3 straight line tests. We’ve plotted the function on the chart. The vertical coloured lines should NOT cross the blue plotted line more than once, if we’re going to call it a function. As you can see, there is only one chart, where a vertical line does not cross the plot more than once. Hence, only the very left example is a function.

Checking it algebraically

We can also work this out algebraically. If we have: x squared + y squared = 1 and we input zero as the x value, we would have: zero squared + y squared = 1, so y squared = 1. The square root of 1 is -1 or + 1 and hence, y has 2 output values and is not a function. This is not the case if x = 2 as the formula would be 4 + y squared = 1. So, y squared = 5 and hence, y = 2.23. As we know though, there must be only one value for y for the entire domain, so this would still not be classed as a function, unless we exclude zero from the domain.

y = x + 1 is a function; y always has a single output whereas, y squared = x + 5 is not a function because, if x = 4, then y squared = 9. Which means, the square root is either 3 or -3.

As a final example, we can say that y = x squared is a function, as it will always have a single output value.

Things to look for are where y is squared or we have the square root of x. Also, we should check what the outcome would be if x were to equal zero.

### Domain & range exclusions

We can make domain and range exclusions with particular notation. Here, we are defining the range as anything greater than or equal to minus 1 and less than or equal to 4. The domain, is greater than or equal to zero & less than or equal to 7.

A function has a dependent variable & an independent variable. in the below examples, I have marked the dependent and independent variables.