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.

✔️Solved: 0

Learning Guide: The Coordinate System

Learning Guide: Introduction to Functions

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

Real-World Analogies

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: $Price = 3 \cdot Weight$ .

Function Notation $f(x)$

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 $0$ : $f(0) = 2(0) + 1 = 1$ .

Equations That Are Not Functions

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

Functions and Graphs

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:

If $f(3) = 5$ , it corresponds to the point $(3, 5)$ on the graph.
If $f(0) = 2$ , it corresponds to the point $(0, 2)$ on the graph.

Distinguishing Functions & the $y$ -axis Crossing

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.

Learning Topics

Frequently Asked Questions

Why does the x-coordinate always come first in an ordered pair?

By mathematical convention, coordinates are always written in alphabetical order as

(x, y)

. This standardized order ensures that anyone around the world can communicate and locate points on a coordinate plane consistently without ambiguity.

What makes a relation a function?

A relation is a function if and only if each input value is associated with exactly one output value. If a single input has multiple different outputs, it is not a function.

How do you find where a function crosses the y-axis?

To find where a function crosses the

y

-axis, calculate

f(0)

by replacing

x

with

0

in the function formula. The resulting point on the graph will be

(0, f(0))

What happens if we input a value outside the domain?

If you input a value outside the domain, the function is undefined for that value. For example, in

f(x) = \frac{1}{x}

, inputting

x = 0

results in division by zero, which has no defined mathematical value.

How can you identify the domain of a function from its graph?

To find the domain from a graph, look at the graph's horizontal extent along the

x

-axis. Find the leftmost and rightmost points of the graph, taking note of whether the endpoints are solid (included) or open circles (excluded).

What is the difference between domain and range?

The domain is the set of all valid input values (usually

x

) that you can feed into a function, while the range is the set of all output values (usually

y

) that the function produces as a result.

How can you identify the range of a function from its graph?

To find the range from a graph, look at the graph's vertical extent along the

y

-axis. Find the lowest and highest points of the graph, taking note of whether these endpoints are included (solid) or excluded (open circles).

What does it mean for a function to be continuous?

Intuitively, it means you can draw the function's graph without lifting your pencil. Formally, a function must be defined at the point, and the graph must connect smoothly without any gaps, jumps, or holes.

How do you find points where a function is not continuous?

Look for inputs that make the function undefined (like division by zero). For example,

f(x) = \frac{x^2 - 16}{x - 4}

is discontinuous at

x = 4

because you cannot divide by zero, creating a hole in the graph.

How do I determine whether a function is increasing or decreasing from its formula?

For linear functions

f(x) = mx + b

, the function increases if the slope

m > 0

and decreases if

m < 0

. For quadratic functions

f(x) = a(x-h)^2 + k

, check the sign of

a

: if

a > 0

, the parabola opens up, so it decreases for

x < h

and increases for

x > h

Why can't I use y-values to define intervals of increase or decrease?

Although we look at the vertical rise or fall (y-values) to determine *what* the graph does, the intervals must specify *where* it happens horizontally. By convention, intervals of increase/decrease partition the domain of the function, which is represented by the

x

-axis.