A function f is a rule for taking one item x, as an input, and producing another item f(x) as an output. Frequently this will involve numbers, so for example the polynomial takes numbers as inputs, and outputs their squares. So in this case, if 1 is input, 1 is output, whilst if 2 is input, 4 is output. However, there are functions that take other objects as inputs. For example, rotation can be thought of as a function that takes a shape as an input, and outputs that shape rotated by a certain number of degrees.
Usually, a function will be written with a formula for the output, in terms of the input. For example, in the case of f above, the formula is . When we want to input a particular value, we insert in in the place of x. So .
If the inputs and outputs are numbers, then the most obvious examples are the polynomials. f(x) = x + 1 is a simple example, a function that adds one to the input. f(1) = 1 + 1 = 2, f(2) = 2 + 1 = 3, and so on. Also, g(x) = 2x is a function that doubles the input. g(3) = 6, g(4) = 8. Another example is h(x) = 2x + 1. This doubles the output, and adds 1. The result is that h(1) = 2*1 + 1 = 3, h(2) = 2*2 + 1 = 5.
i(x) = x is another function that does nothing to x. So i(1) = 1, i(2) = 2, and so on. This concept will be important later on.
Some functions do not have numbers as outputs. For example, we can define a function pres that takes a number n, and returns the name of the nth American President. So, pres(1) = George Washington (the first president), pres(2) = John Adams (the second president), and so on.
Similarly, not all functions take numbers as inputs. For example, we can define a function pres2 that takes the name of an American President (the nth in sequence), and outputs n. Thus pres2(George Washington) = 1, pres2(John Adams) = 2. This is the reverse of pres defined above.
We can take 2 functions, and combine them to form another function, by applying them one after another. This is called composition. For example, take f(x) = x + 1, and g(x) = 2x. We can apply g to x, and then f to the output g(x), to get f o g, which is the function we get by combining the functions f and g in this way. We can think of this as like a factory production line, where the output of one process or machine becomes the input of the next. Here, the output of g becomes the input of the function f.
In our example, (f o g)(x) = f(g(x)) = f(2x) = 2x + 1. So the combined function f o g is in fact h, which we met earlier. To test this, we can take (f o g) applied to 8. g(8) = 2*8 = 16. Then we apply f to 16, to get f(16) = 16 + 1 = 17. 17 = 2*8 + 1, as expected.
We can also apply the functions in the opposite order to get g o f, where we apply f first then g. f(x) = x + 1, and g(x+1) = 2(x+1) = 2x + 2. So (g o f)(x) = 2x + 2. So, the order in which we combine the functions makes a difference to the final result. Again, we test this formula by applying the function to 8. f(8) = 9, and g(9) 18. 18 = 2*8 + 2, as expected.