Introduction to Functions

Learning Objectives

Define a function and its components (input, output, rule).

Use function notation, f(x).

Distinguish between the independent and dependent variables in a function.

Use the vertical line test to determine if a graph represents a function.

The Function Machine

In mathematics, a function is a special relationship between a set of inputs and a set of outputs. It's like a machine: you put something in, the machine follows a specific rule, and it gives you a single, predictable thing out.

The Rule of a Function

The most important part of the definition of a function is this:

For each input, there is exactly one output.

Think of a vending machine. If you press the button for 'soda' (the input), you expect to get a soda (the output). You wouldn't be happy if you sometimes got soda, sometimes got chips, and sometimes got nothing! That would not be a functioning machine.

Input: The value that goes into the function. It is also called the independent variable. We usually represent it with x.

Rule: The operation or set of operations that the function performs on the input.

Output: The value that comes out of the function. It is also called the dependent variable because its value depends on the input. We usually represent it with y.

Function Notation

We often use a special notation for functions. Instead of writing `y = 2x + 1`, we can write:

f(x) = 2x + 1

This is read as 'f of x equals 2x plus 1'.

f is just the name of the function (we could also use g(x) or h(x)).

(x) shows that 'x' is the input variable.

f(x) represents the entire output value (it's another name for y).

This notation is useful because it's a shorthand way to ask 'what is the output when the input is a certain number?'

Example: For the function `f(x) = 2x + 1`, find `f(3)`.

This means 'substitute 3 for x in the rule'.

`f(3) = 2(3) + 1`

`f(3) = 6 + 1`

`f(3) = 7`

So, when the input is 3, the output is 7. This corresponds to the ordered pair (3, 7).

The Vertical Line Test

A simple way to tell if a graph represents a function is to use the vertical line test.

The Test: If you can draw any vertical line on the graph that crosses the graph in more than one spot, then the graph is NOT a function.

Why it works: A vertical line represents a single x-value (a single input). If it hits the graph in two places, it means that one input (x) has two different outputs (y), which violates the definition of a function.

A circle, for example, is not a function because a vertical line can cross it in two places. A straight line or a parabola, however, are functions.

Key Terms

**Function: A relationship between a set of inputs and a set of outputs where each input is related to exactly one output.
**Input: The value put into a function. Also known as the independent variable (x).
**Output: The value that results from a function. Also known as the dependent variable (y).
**Function Notation: A way of writing a function, such as f(x) = 2x + 1, where f(x) represents the output for a given input x.
**Independent Variable: The input of a function (x), which can be changed.
**Dependent Variable: The output of a function (y), whose value depends on the input.
**Vertical Line Test: A test used to determine whether a graph represents a function. If any vertical line intersects the graph more than once, it is not a function.