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\(f(x)\)

Function
Rules

Learn to interpret and write function rules in words and in equation form: discover the pattern that turns every input into an output.

Pre-Algebra

What Is a Function?

A function is a rule that takes every input value and produces exactly one output. Think of it as a reliable machine: you feed it a number and it always gives back the same answer.

The Function Machine

Rule
\( \times 3 + 1 \)
 Input 4 Output 13

Why Do Function Rules Matter?

Predict
A rule lets you find any output without listing every pair: just plug in!
Communicate
Equations like \(y = 2x + 5\) say in one line what a whole table says in rows.
Connect
Function rules link tables, graphs, and real-world patterns together.

Reading an Input-Output Table

Input \(x\) Output \(y\)
15
28
311
414
517

From Words to Equations

A function rule in words says "multiply the input by 3, then add 2." To write it as an equation, replace the words with symbols:

Multiply by 3, then add 2
\( y = 3x + 2 \)

Expression vs. Equation vs. Function Rule

Concept Example Key Feature
Expression \(3x + 2\) No equals sign
Equation \(3x + 2 = 14\) Equals sign, solve for \(x\)
Function Rule \(y = 3x + 2\) Shows output for every input

The General Form

\( y = mx + b \)
\(m\)
The multiplier (rate of change): how much \(y\) changes when \(x\) goes up by 1.
\(b\)
The starting value (constant): the output when \(x = 0\).

How to Find the Rule

1
Look at the change. Find how much \(y\) increases each time \(x\) increases by 1. That number is \(m\).
2
Find the start. Plug \(x = 0\) into the table, or use \(b = y - mx\) with any row. That number is \(b\).
3
Write the rule. Combine: \(y = mx + b\).
4
Check. Plug in a different pair from the table. If both sides match, your rule is correct.

Example 1: Finding the Rule

\(x\) \(y\)
04
17
210
313

Example 2: No Zero Row

\(x\) \(y\)
29
313
417
521

Example 3: Negative Start

\(x\) \(y\)
13
28
313
418

Seeing the Rule on a Graph

0 1 2 3 4 5 0 2 4 6 8 10
\(b = 1\)
 +1 +2
\(y = 2x + 1\)

Writing Rules in Words

Sometimes you need to describe a function rule in a sentence rather than using symbols. Translate each part of the equation:

1
\(y = 3x + 4\) : "Multiply the input by 3, then add 4."
2
\(y = x - 7\) : "Subtract 7 from the input."
3
\(y = \frac{x}{2} + 3\) : "Divide the input by 2, then add 3."

Common Mistakes

✕ Confusing \(m\) and \(b\)
Writing the constant jump as the starting value and vice versa. Remember: \(m\) is the change, \(b\) is the value when \(x = 0\).
✕ Forgetting Negative \(b\)
If \(b\) turns out negative, write \(y = 5x - 2\), not \(y = 5x + 2\). Always check the sign.
✕ Skipping the Check
Not plugging a second pair back in. One wrong arithmetic step can make your entire rule incorrect.

Summary at a Glance

Representation Example When to Use
Table Rows of \((x, y)\) pairs Listing specific values
Words "Multiply by 3, add 4" Explaining to someone
Equation \(y = 3x + 4\) Quick calculation
Graph Straight line Visualizing the pattern

Question 1 of 5

A table shows: when \(x = 1\), \(y = 6\); when \(x = 2\), \(y = 10\); when \(x = 3\), \(y = 14\). What is the function rule?

Question 2 of 5

Which sentence describes the rule \(y = 5x - 3\)?

Question 3 of 5

Using the rule \(y = 2x + 7\), what is the output when \(x = 10\)?

Question 4 of 5

A function rule says: "Divide the input by 3, then add 5." Which equation matches?

Question 5 of 5

In the rule \(y = mx + b\), what does \(b\) represent?