One of the nice consequences of requiring that there be only one output for each input is that we can write the function in a special way. For example, you know about the xy or Cartesian coordinate system, so you already know that you write an equation for a parabola as \(y=x^2\). Notice that the input value, also called the independent variable x and output value y, called the dependent variable since it's value depends on x are written on separate sides of the equal sign. Notice also that there is one and only one y by itself. This is significant in that only functions can be written like this, i.e. with the equation solved for y.

Since we write the y by itself on one side of the equal sign, we can use a special type of notation, which we call function notation. This allows us to drop the y and replace it with a 'name'. This name is most often just a single letter but it can also be longer.

For example, we can rewrite the parabola equation \(y=x^2\) as the function \(f(x)=x^2\). Then we can refer to this parabola equation as the function f(x) or just f. The special notation on the left side of this equation uses parentheses but in this case we are not multiplying f by x. We are saying that the function f has independent variable x. We usually say 'f of x' when referring to this function.

Another nice thing is that we are not tied in to always using f and x when describing this function. The following functions are all equivalent since we always get the same output value associated with each input value.

In order to find out what is going on with a function, we will often need to put in values and see what we get out. The notation \(g(x)=x^2\) makes it really easy to see what we need to do. For example, if we need to know that output value for an input value of 3, we would write \(g(3)\). Comparing this to the function \(g(x)=x^2\), it should give you a clue that we take the input value of 3 and put it everywhere there is an x in the function. So, \(g(3)=3^2=9\), i.e. \(g(3)=9\). It's as simple as that.

If there are more x's in the function, you just replace ALL of the x's with the number. For example, if we have a function \(h(t)=t^2+t-6\), then \(h(3)=3^2+3-6=6 ~~ \to h(3)=6\). That's it.

Let's take a minute to watch a quick video that explains this notation and shows more examples. Toward the end of the video he talks about composition of functions, which we discuss next.

How to Read and Do Proofs: An Introduction to Mathematical Thought Processes

Khan Academy - Introduction to functions [9min-33secs]

Khan Academy - Functions (Part 2) [9min-53secs]

Khan Academy - Functions (Part 3) [9min-14secs]

Khan Academy - Functions (Part 4) [6min-7secs]