Functions: Introduction to Functions

Understand what a function is, explore real-world analogies, master function notation f(x), and learn to identify points and y-intercepts.

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Learning Guide: The Coordinate System

A function is a mathematical rule or relationship that connects inputs to outputs. For every input, a function assigns exactly one output.

Vending Machine: You input a code (like $A1$ ), and the machine outputs exactly one specific snack (like potato chips). If one code could output different items at random, it wouldn't work like a function!
Temperature Over Time: For any specific hour of the day (input), there is exactly one specific temperature (output).
Buying Apples: If apples cost $3$ dollars per kilogram, the total price (output) depends on the weight (input). We can write this rule as: $Price = 3 \cdot Weight$ .

We write functions using the notation:

f(x) = \text{expression}

$f$ is the name of the function.
$x$ is the input variable.
$f(x)$ represents the output value (the result of the rule applied to $x$ ).

For example, if $f(x) = 2x + 1$ :
If we input $3$ , we substitute $x$ with $3$ : $f(3) = 2(3) + 1 = 7$ . The input $3$ gives the output $7$ .
If we input $0$ , we substitute $x$ with $0$ : $f(0) = 2(0) + 1 = 1$ .

It is important to understand that there are valid mathematical equations that are not functions. One of the simplest examples is

x = 1

(or

x = c

for any constant

c

). While

x = 1

is a perfectly valid and meaningful mathematical equation (describing a vertical line), it is not a function because the input

x = 1

corresponds to infinitely many output values

y

. The validity of a mathematical equation should not be confused with whether it defines a function.

Figure 1b: The vertical line

x = 1

is a valid mathematical equation but not a function, because the input

x = 1

corresponds to multiple different output values (like

y = 2

and

y = -2

When we draw a function on a coordinate system, the input (

x

) corresponds to the horizontal position, and the output (

f(x)

y

) corresponds to the vertical position. Each input-output pair gives us a point on the graph:

We can often distinguish functions by looking at where they cross the vertical

y

-axis. The point where a function crosses the

y

-axis is called the y-intercept. It is always found by setting the input to

0

, which gives the point

(0, f(0))

.

For example, look at the two lines in the graph below:

The blue line representing $f(x) = x + 1$ crosses the $y$ -axis at $(0, 1)$ , because $f(0) = 1$ .
The red line representing $g(x) = -x + 3$ crosses the $y$ -axis at $(0, 3)$ , because $g(0) = 3$ .
Since $f(0) \neq g(0)$ , they cross the vertical axis at different heights, helping us tell them apart!

Figure 1: Two functions f(x) = x + 1 (blue) and g(x) = -x + 3 (red), crossing the y-axis at f(0) = 1 and g(0) = 3 respectively.

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