Introduction to Differential Calculus

The Mathematical Function

Calculus is all about functions.
What is a function you ask?

Let's start with mathematical relations. A relation relates one value to another, using a set of operations. Operations such as division, multiplication, subtraction, addition, square roots, cube roots, log, sin, cos, tan, and exponentiation. But seeing as this is about functions, I will not focus on relations.

A function is a type of relation that relates an input to an output using operations too, where one input has only one output. Mathematically speaking, the input is usually called the x - value whilst the output is the y - value. This is the standard used in mathematics as it corresponds to the axes on the Cartesian plan which allows a function to be graphed. Note that, with functions, only one y - value is permissible for every x -value but there can be two x - values for every y - value.

The Cartesian plane:

- y axis for the vertical axis             - x axis for the horizontal axis

Another notation for the function is the term f(x).
f(x) denotes the function that the x value has to go through in order to get the output.
For example, lets take a look at the function f(x) = x² + 1
Let's plug an input value in for x, such as x = 2

If that's the case, we get     f(x) = (2)² + 1
And hence,                                f(x) = 5  

Now, we know that f(x) defines the function, but if we want to graph the function we need an output value, so we use 'y' when we graph functions. Thus, to get the equation of the function we just make y equal f(x), giving us  y = f(x)

Using the same function as above    y = x² + 1
And hence,                                                   y = 5

We get one definite output for one input. If we graph all the points of inputs and outputs, by corresponding them with the x and y axis, we get a graph.

This is the graph of y = x² + 1

The black lines on the graph above goes to show that our value of x being 2, has given us a y output of y being 5. Therefore, the point (2,5) lies on the function y = x² + 1 or f(x) = x² + 1
(We use the latter when referring to the function in general and the first one when referring to the the equation of the function)

By making y equal f(x) we get the equation of the function.
The equation of the function can be anything. The one used in the example above was a "quadratic equation" where the highest power of x is 2. The graph results in a parabola.
Here are some other common equations:

y = sin(x) 

y = x³

y = log(x)

There are endless possibilities to what a function can be and they can get very complex.

Functions, and a general understanding of basic algebra is needed in order to start on calculus. This chapter only provides a refresher on functions, as in further chapters, it will be assumed knowledge.