What are functions in Mathematics? In this post, let’s talk more about the topic, functions.

## How Functions are Denoted in Math

We denote a function, f, in terms of x as f(x). Think of it as writing y = x +3. y is a function in x, and we can replace it with f(x). Apart of using f for functions, g and h are also very often used for functions.

E.g. h(x) = x-3

g(x) = ln(x+5)

f(x) = sinx -2

Of course, we can also use other letters to denote functions. E.g. s(x) = x+lnx. Also, functions do not need to be in terms of x. They can be in terms of any other variables that you like. For example, u is a function in terms of t: u(t) = 2t-7sint.

## Math Definition of Functions

The Math definition of functions has 2 parts in it. The 2 parts of the function:

- shows a relationship between an input and an output AND
- each input gives a single output only

### Relationship between an input and an output

Let’s explore the first part of function, i.e. a function shows a relationship between an input and an output. For example, let’s use f(x) to define a relationship between x and f(x).

E.g. f(x) = 3x+2

We call 3x+2 the rule of f. Based on the above relationship, we know that when given any input, the function, f, will first multiply the input (or x) by 3, and then add 2 to it. Hence, this is the relationship. If x =2, then f(2) = 3(2) +2 = 8. If we were to replace x by z, then f(z) = 3z+2. Similarly, we can replace x by 2x, and we get f(2x) = 3(2x) +2 =6x+2.

### Each input gives a single output

For each “x” or input, we get only one output value (i.e. a single value for f(x)).

f(x) = 3x + 2 is a function, because it only gives a single value of f for any value of x.

However, if we have a function g(x), and g(x) = ±(x-2). g(x) is not a function since it gives more than 1 value for each input. For instance, g(1) = 1 or -1.

Note that each input gives only a single output, however, you can have the same output for multiple inputs. E.g. in f(x) = x^{2} , both f(1) and f(-1) gives a value of 1. This means that to get a value of 1 for f(x), x or the input can be 1 or -1.

## Domain and range of functions

In H2 Math, domain should be defined together with a function. The domain of the function is the values that the inputs can take. If we have a function in terms of x (e.g. f(x)), then the domain refers to the values that x takes.

The range is the values that the output of a functions take. If we have a function, f, then the range refers to the values that f can take, given the domain. The range of a function can be obtained by plotting the graph, and looking at the range of values that y can take for the given domain.

## One-One Function

A one-one function is one where each output maps to a single input.

To test if a function, f(x) is one-one function, plot y= f(x) over its domain. A horizontal line y= k where k is any value in the range of f, cuts y = f(x) at one and only- one point.

## Inverse Function

The inverse function of f(x) is f^{-1}(x). f^{-1}(x) will map the output of f to the input of f.

In order for an inverse function to exist, it must be a one- one function.

### Useful results between functions and inverse functions

range of f = domain of f^{-1}

domain of f = range of f^{-1}

ff^{-1} (x) =f^{-1}f(x) = x

Graph of y= f^{-1}(x) can be obtained by reflecting y=f(x) about the line y= x.

## Composite Functions

Composite functions are functions within a function. E.g. fg(x), hg(x), f^{2}(x), etc.

For instance, we can think of fg(x) as f(g(x)), where g(x) is the input of the function f, and g(x). Such functions are called composite functions.

For a function gf(x), for it to exist, the range of f must be equal to or a subset of the domain of g (i.e. range of f ⊆ domain of g). If gf exists, then the domain of gf = domain of f.