# Functions

By Murray Bourne, 22 Feb 2010

A lot of people have difficulty with **functions** in math. I suspect it's because function notation is not very learner-friendly.

Yousuf, one of my regular correspondents, got stuck on the following problem recently.

**What is the area of the rectangle ADEB shown in the diagram?**

The curve is the graph of *y* = 1/*x*^{2} (for positive *x*), and *r* is some arbitrary value of *x*.

We'll come back to this question a little later. I suspect his problem with this question was due to a rusty conceptual understanding of functions.

## Functions Overview

A function is simply an expression involving variable(s).

We usually write a function of the variable *x* using the notation: *f*(*x*). A function has at most 1 value for each value of *x*.

For example, if *f*(*x*) = 5*x*^{2} + 3, we can find the value of the function if we choose *x *= 0 as follows.

*f*(0) = 5(0)^{2} + 3 = 5 × 0 + 3 = 3

Now, this is a good example of the **notation** problem I was talking about at the beginning. We write "*f*(0)" (*f* bracket 0 bracket) to mean "evaluate the function expression by substuting 0 every time we see an *x*" and we see this on the left hand side of this equation.

But on the right hand side, I have written "5(0)^{2}" (5 bracket 0 bracket squared) and this means "5 × 0^{2}". We need to be careful with this - writing 2 different concepts with what is essentially the same notation.

It is a shame that function notation is so clumsy and causes problems for newbies.

Let's look at some more examples for our function *f*(*x*) = 5*x*^{2} + 3.

*f*(2) = 5(2)^{2} + 3 = 5 × 4 + 3 = 23.

*f*(10) = 5(10)^{2} + 3 = 5 × 100 + 3 = 503.

If we were to substitute many more values of *x* and plot the dots on a graph, we would get the following:

**Note: **On the vertical axis, I put *f*(*x*), but I could have also put "*y*", since the convention in math is the vertical axis represents the function value. Often you'll see it written *y = f*(*x*).

OK so far?

Now, let's make things a bit more interesting. What is *f*(*a*)? We just substitute *a* everywhere there is an *x* in the original function, like we did before with the numbers:

*f*(*a*) = 5(*a*)^{2} + 3 = 5*a*^{2} + 3

Let's do another. In this next case, *f*(*a* + 4), we are just replacing each *x* in the original function expression with *a* + 4.

*f*(*a* + 4) = 5(*a* + 4)^{2} + 3 = 5(*a*^{2} + 8*a* + 16) + 3 = 5*a*^{2} + 40*a* + 83

Of course, we need to be careful to expand out the brackets properly!

## A Different Function

Let's change our function to .

This is the curve we met in the question at the the beginning of this article.

If *x* = 1*, *we replace every *x* in our expression with 1 and we have:

What *f*(1) means on a graph is the distance from the *x-*axis to the graph is 1 unit. The function value is the **height** of the graph for that *x*-value.

Now let's do *f*(3*a*).

The value

represents the height of the graph when *x* = 3*a.* We need to be careful with the brackets.

## Back to Our Problem

Here's the graph again:

So how do we find the area of the rectangle BADE? The **width **of the rectangle is quite straightforward, as the distance from *r* to *r* − 1 is just 1 unit. But we need to find the height AD.

AD is just the function value *f*(*r*):

So the area of the rectangle is just

Area = width × height =

## What if we needed the height BC?

We would just find the function value as follows.

BC =

## Functions of 2 Variables

The functions above only have one variable and they describe a curve in 2-D space.

To describe a 3-D surface, we need to use 2 variables.

We write a function of 2 variables using this notation:

*z = f*(*x,y*)

The "*z*" indicates the height of the surface for particular values of *x* and *y*.

An example of a 3-dimensional surface is *z* = *x*^{2} + 3 sin* y.*

## More Information

See this chapter for a lot more examples of functions: Functions and Graphs. (2 dimensional)

This is an introduction to 3-dimensional Coordinate System.

See also Towards more meaningful math notation where I suggest an alternative to the current confusion.

See the 14 Comments below.