Introduction to Functions

In earlier parts of the course, you have worked with numbers, variables, and algebraic expressions. You have also seen equations that link variables together. This chapter introduces a new viewpoint: instead of looking only at single equations, we look at relationships between inputs and outputs. These relationships are called functions.

The main goals here are:

• To understand what a function is, in simple, concrete terms.
• To see different ways a function can be described.
• To learn how to think of one quantity as depending on another.

Later chapters will formalize this further and introduce function notation. Here we build the basic intuition and use simple examples.

Inputs and outputs: machines and rules

A helpful way to think about a function is as a machine or rule that:

• takes an input (usually a number),
• does a consistent operation to it, and
• produces exactly one output.

You can picture it like this (mentally, or on paper):

• You choose a number and place it into the “input” slot.
• The function applies some rule, such as “multiply by 2” or “add 5.”
• A new number comes out of the “output” slot.

For example, suppose the rule is “multiply the input by 3 and then subtract 1.”
If the input is $2$, the output is:
$$
3 \cdot 2 - 1 = 6 - 1 = 5.
$$

If the input is $4$, the output is:
$$
3 \cdot 4 - 1 = 12 - 1 = 11.
$$

The same rule is applied to every input, and each input has one output.

This point is important:

• A function must give exactly one output for each allowed input.
• Different inputs can give the same output; that is allowed.
• But the same input cannot give two different outputs.

Later you will see this expressed in more formal ways. For now, think:
“Each input number has one assigned output number according to a rule.”

Examples of simple function rules

Here are some simple rules that define functions:

1. “Add 5 to the input.”
• Input: $2 \rightarrow$ Output: $2 + 5 = 7$
• Input: $-3 \rightarrow$ Output: $-3 + 5 = 2$
2. “Double the input.”
• Input: $4 \rightarrow$ Output: $2 \cdot 4 = 8$
• Input: $-1 \rightarrow$ Output: $2 \cdot (-1) = -2$
3. “Take the input, multiply by 4, then add 3.”
• Input: $1 \rightarrow$ Output: $4 \cdot 1 + 3 = 7$
• Input: $0 \rightarrow$ Output: $4 \cdot 0 + 3 = 3$

Each of these rules describes a function. You do not yet need to use special function notation; that will be covered later. For now, focus on the idea: one input, one output, same rule every time.

Checking if a rule describes a function

To decide if a rule describes a function, ask:

1. Is the rule clear enough that for any allowed input, you can find a single output?
2. Does each input lead to exactly one output (no confusion or choice)?

Consider these examples:

• Rule A: “Square the input” (multiply the number by itself).
• Input: $2 \rightarrow 4$
• Input: $-2 \rightarrow 4$
• Each input gives exactly one output. This is a function.
• Rule B: “Take the input and give the number of its square roots.”
• Input: $4 \rightarrow$ The square roots are $2$ and $-2$.
• If you tried to use “take a square root” in a way that picks both $2$ and $-2$ as outputs for input $4$, that would not fit the idea of a function, because one input is giving two outputs.
• If, on the other hand, you clearly define the rule as “take the positive square root of the input,” then each input has exactly one output, and it is a function.
• Rule C: “Assign to each person their current favorite number.”
• A person might not have a single, well-defined “favorite number” or might change their mind. This makes the rule unclear.
• For a function, the rule must be clear and fixed.

The key test: for every allowed input, is there exactly one output, determined without ambiguity?

Different ways to represent a function

Even for beginners, it is useful to see that the same function can appear in several forms. For now, we use three basic representations:

1. Words (a verbal description).
2. A table of values.
3. A graph (as points on the coordinate plane, which you will study in more detail later).

1. Verbal description

A verbal description uses ordinary language to explain how to get from input to output. For example:

• “Take the input, multiply by 2, and then subtract 3.”
• “Take the number of hours worked and multiply by 10 to get pay in dollars.”

A verbal description is the most natural way to start thinking about functions; it mirrors how real-world situations are described.

2. Table of values

A table lists some inputs and the corresponding outputs. For example, for the rule “multiply by 2 and add 1,” you might have:

Input (number) | Output (number)
-------------- | ---------------
$-2$ | $-3$
$-1$ | $-1$
$0$ | $1$
$1$ | $3$
$2$ | $5$

Each row shows a pair: input and its matching output.

Important points:

• A table usually shows only some of the possible input–output pairs, not all of them.
• The pattern in the table comes from the function rule.
• In a function, each input appears at most once in the input column, because each input has only one output.

If an input appears twice with different outputs (for example, input $1$ matched with both $2$ and $3$), then the table does not describe a function.

3. Graph as plotted points

Later, in the coordinate plane chapters, you will learn to plot points of the form $(x, y)$. For now, it is enough to know:

• For a function, you can think of the input as $x$ and the output as $y$.
• Each input–output pair becomes a point $(x, y)$ on a graph.

Example: Suppose the rule is “double the input.” Some input–output pairs are:

• Input $-2 \rightarrow$ Output $-4$ gives the point $(-2, -4)$.
• Input $0 \rightarrow$ Output $0$ gives the point $(0, 0)$.
• Input $3 \rightarrow$ Output $6$ gives the point $(3, 6)$.

If you plotted several such points, you would begin to see a pattern (in this case, they line up in a straight line, which is studied later as linear functions).

You do not need to graph functions in detail yet. Just understand that:

• A graph is a picture of the set of input–output pairs.
• For a function, there is only one $y$-value for each $x$-value that appears.

Functions in everyday situations

Many everyday relationships can be thought of as functions, even if people do not use that word.

Example: Total cost as a function of quantity

Suppose apples cost \$2 per apple. The total cost depends on how many apples you buy.

Rule: “Multiply the number of apples by 2 (dollars).”

Some values:

Number of apples (input) | Total cost in dollars (output)
-------------------------|--------------------------------
$0$ | $0$
$1$ | $2$
$2$ | $4$
$3$ | $6$

You can say: “Total cost is a function of the number of apples.” If you know how many apples, you can find the cost by the same rule every time.

Example: Distance traveled at constant speed

If you walk at a constant speed, the distance you travel depends on how long you walk.

Suppose you walk at $5$ kilometers per hour (km/h). Then:

Rule: “Multiply the number of hours by 5 to get distance in kilometers.”

Time in hours (input) | Distance in km (output)
----------------------|------------------------
$0$ | $0$
$1$ | $5$
$2$ | $10$
$3$ | $15$

Here, distance is a function of time (under the assumption of constant speed).

Example: Temperature conversion

To convert degrees Celsius to degrees Fahrenheit, there is a rule:

“Multiply the Celsius temperature by 9, divide by 5, then add 32.”

This rule gives a function from Celsius (input) to Fahrenheit (output). Each Celsius value leads to exactly one Fahrenheit value.

You do not need to memorize this rule here. The point is that many familiar conversions are functions.

Input and output variables

In earlier chapters, you saw variables like $x$ and $y$ used in algebraic expressions and simple equations. With functions, we usually use:

• One variable for the input (often $x$).
• Another variable for the output (often $y$).

In that language, we might say:

• “$y$ is a function of $x$.”
• “The value of $y$ depends on the value of $x$.”

This means:

• Once you choose $x$, the rule tells you what $y$ must be.
• The variable $x$ is sometimes called the independent variable (you can choose it freely from allowed values).
• The variable $y$ is sometimes called the dependent variable (its value depends on $x$).

Example: “Multiply the input by 3 and add 2.”

• If we use $x$ as input and $y$ as output, then the rule can be described in words:
• “To find $y$, take $x$, multiply by 3, and then add 2.”
• You can make a table:

$x$ (input) | $y$ (output)
------------|-------------
$-1$ | $-1$
$0$ | $2$
$1$ | $5$
$2$ | $8$

Later, you will learn function notation that replaces “$y$” with more specific symbols, but the basic idea remains the same: one variable depends on another through a rule.

When a relationship is not a function

Sometimes two quantities are related, but not in a way that defines a function. This happens when a single input could lead to more than one possible output.

Here are some simple examples in words and tables.

Example 1: Grade to possible test scores

Imagine the rule: “Match a letter grade (A, B, C, ...) to a test score that could have produced it.”

One letter grade usually corresponds to many possible scores. For example:

Letter grade (input) | Possible scores (output)
----------------------|-------------------------
A | $90, 91, 92, \dots, 100$
B | $80, 81, 82, \dots, 89$

Here each input (a letter grade) has many outputs (many scores). This is not a function from letter grades to single scores.

Example 2: Person to favorite color (not clearly defined)

If the rule is: “Assign to each person their favorite color,” there are problems:

• A person might like several colors equally.
• Their favorite might change.
• The rule is not clear and fixed.

Because a function must assign exactly one output to each input using a clear, consistent rule, this kind of vague situation is not a good function example.

When you are given a table or verbal description, ask:

• For each input, is there exactly one output?
  If not, the relationship is not a function in the sense used in algebra.

Summary

• A function is a rule that assigns exactly one output to each allowed input.
• You can think of a function as a machine: you put in an input, apply the rule, and get an output.
• Common ways to represent functions at this level:
• A verbal description (“multiply by 2 and add 3”),
• A table of input–output pairs,
• A graph made of points $(x, y)$, where $x$ is input and $y$ is output.
• In functional relationships, one quantity (the output) depends on another (the input). We often call the input variable $x$ and the output variable $y$.
• A relationship is not a function if some input has more than one possible output or if the rule is not clearly defined.

Later chapters will introduce specific function notation and connect these ideas more formally to the coordinate plane and graphs.