I would like to start the journey to explain calculus. This journey actually started with algebra and trigonometry. Calculus is heavily into functions so a review of functions and the associated notation is timely.
Instead of relating a variable with another using the common variables x and y where y is all by itself in the left side (eg y = 3x² -7x + 2), we can replace y with f(x): y = f(x) = 3x² -7x + 2. This notation highlights the fact that x is an independent variable (that is, we are free to choose a valid value of x), but once we do, y or its new notation f(x), depends on the value of x that was chosen. So y is the dependent variable.
This notation also results in simplifying desired operations. Instead of saying “Evaluate y when x = 3″, you can just write f(3). So given a definition of a function, you replace the independent variable with whatever is in the brackets:
f(x) = 3x² -7x + 2
f(3) = 3(3)² -7(3) + 2 = 8
The thing in the brackets does not have to be a number. It can be another variable or even an algebraic expression:
f(x) = 3x² -7x + 2
f(a) = 3a² -7a + 2
f(x+h) = 3(x+h)² -7(x+h) + 2
You can even have functions of functions:
f(x) = 3x² – 5x + 1
g(x) = x² – 7
f[g(x)] = f(x² – 7) = 3(x² – 7)² – 5(x² – 7) + 1
g[f(x)] = g(3x² – 5x + 1) = (3x² – 5x + 1)² – 7
The domain of a function is all the valid values of x that can be used. Many times, the domain of a function (like f(x) and g(x) above) is just any real number. But there are functions where you cannot use just any number. For example, consider
\[ {f}{(}{x}{)}\hspace{0.33em}{=}\hspace{0.33em}\frac{3}{{x}{-}{2}} \]There is one value of x you cannot use. That value is 2 because that will make the denominator 0, and as you know, this will bring the maths police to your door. So the domain of this function is all real numbers except for 0.
Now consider
\[ {f}{(}{x}{)}\hspace{0.33em}{=}\hspace{0.33em}\sqrt{{x}{-}{2}} \]Another illegal operation is taking the square root of a negative number. The requirement for this function is that x – 2 has to be 0 or greater. For this to be true, x must be greater than or equal to 2. The phrase ” greater than or equal to” can be replaced by the maths symbol ≥. So the domain of this function is x ≥ 2.
There are other reasons why the domain of a function is restricted, but the most common things to look for is dividing by 0 or taking the square root (or any even root) of a negative number.
I will be using functional notation in the following posts extensively. You will become very familiar and comfortable with it.