Functional Notation, Part 1

Any maths beyond algebra relies on something called functional notation. I used this in one of my posts on Newton’s Laws but more needs to be said if you are to be comfortable with it.

We have looked at many equations to date. Most involved x and y. For example,

y = 3x² – 5x + 1

This will plot as a parabola on an xy coordinate system. The plot is a picture of all (x, y) pairs of numbers that, when substituted in the above equation, will result in a true statement. For example, (0, 1) would be a point on that parabola because 1 = 3(0)² – 5(0) + 1 = 1.

Most of the equations we have looked at have y on the left side and all the other things with x and stand-alone numbers are on the right side. In this form, it is easy to choose a number to replace x with, then do the maths with that number on the right side to find the corresponding y to make the equation true. For example, let’s choose “1” in place of x. The corresponding y will be

y = 3x² – 5x + 1 = 3(1)² – 5(1) + 1 = 3 – 5 + 1 = -1

So (1, -1) is also a point on this equation’s plot.

This is frequently done: choose a value for x, then replace x with that value and do the maths on the right side and find the corresponding y. Notice that we are free to choose a value for x, but once we do, the value for the corresponding y is fixed. For that reason, x is called the independent variable and y is called the dependent variable: y depends on the x we choose.

If y depends on the x we choose, then another way to say this is that “y is a function of x“. The new functional notation makes use of this by replacing y with f(x), read “function of x“. So the functional form of the equation above is

f(x) = 3x² – 5x + 1

This will plot exactly the same but we would replace the y label on the vertical axis with f(x):

So now it is easy to ask the question “What is the value of the function if x = 0″ by just replacing the “x” in f(x) with “0”, that is, f(0):

f(0) = 3(0)² – 5(0) + 1 = 1

So, f(0) = 1. We saw above that f(1) = -1. So now you will see the general coordinate as (x, f(x)) instead of (x, y). This is just a difference in notation – the plot stays the same.

There are some properties of functions and a few more definitions that will be explored next time.