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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/x2 (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) = 5x2 + 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 × 02". 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) = 5x2 + 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:

x^2 + 3

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 = 5a2 + 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(a2 + 8a + 16) + 3 = 5a2 + 40a + 83

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

A Different Function

Let's change our function to 1/(x^2) .

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(3a).


The value

1 / 9a^2

represents the height of the graph when x = 3a. 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 = 1/r^2

What if we needed the height BC?

We would just find the function value as follows.

BC = f(r - 1)

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 = x2 + 3 sin y.

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.

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