# What's a function?

##### 2 Answers

Please see below.

#### Explanation:

A function is a rule which maps values of a variable into another variable. The first variable is the input, which when put in the **rule**, (in mathematics, this rule is defined by **a mathematical formula** ) results in another variable, which is the output.

These rules are generally described as say

where **input value** **rule** **output**

For example let

if we have input

if

and if

Hence, for different values of **input**, we get different values of **output**, which is decided by the **rule or the matemaical formula**.

The **input variable** is called **independent variable** and the **output variable** is called the **dependent variable**.

The values that input variable or independent variable can take is called **domain** and values that output variable or dependent variable can take is called **range**.

Generally names of **variables** are represented by alphabets such as **rules or formulas** are represented by

The **rules or formulas** could be simple linear or quadratic functions, polynomials, algebraic functions (more like fractions with polynomials as numerators and denominators), trigonometric functions, exponential functions etc.

A function is a rule we apply to an input and which produces a single result.

#### Explanation:

**A function is simply a rule that we apply to an input** (could be a number, could be a variable, could even be another function).

Let's look at how functions are written:

So let's say that the rule is

Functions can get more and more complicated, as can the values that are inputted. But the basic rule still applies - input a value, get a value out.

There is one thing to note about the value in/value out thing - **we want a single value to come out from any value we put in.** If we get more than one value, the rule suddenly isn't a rule - it's more of a list of possibilities. And so if we get more than one value, it isn't a function.

This is where the **vertical line test** comes in - if we graph our function (which we can do - the

Here's our original example

graph{x+2 [-10.04, 9.96, -3.16, 6.84]}

I can draw vertical lines and encounter only one value:

graph{(y-(x+2))(y-1000x+1000)=0 [-10.04, 9.96, -3.16, 6.84]}

But if I have a graph that looks like this:

graph{(y-sqrtx)(y+sqrtx)=0 [-5, 9.96, -5, 6.84]}

The vertical line test will fail - and therefore the graph isn't a function:

graph{(y-sqrtx)(y+sqrtx)(y-1000x+1000)=0 [-5, 9.96, -5, 6.84]}