Function Rule — Definition, Formula & Examples
A function rule is the mathematical expression or equation that describes how to turn each input into its output. For example, the function rule tells you to multiply the input by 2, then add 3.
A function rule is an algebraic formula that defines the relationship between the independent variable (input) and the dependent variable (output) of a function, allowing the output to be computed for every value in the domain.
Key Formula
Where:
- = The input value (independent variable)
- = The output value (dependent variable)
- = The function rule applied to x
How It Works
You use a function rule by substituting an input value into the expression and simplifying to find the output. If someone gives you a function table, you look for the pattern that connects every input to its output — that pattern is the function rule. Once you know the rule, you can predict the output for any input, even values not listed in the table.
Worked Example
Problem: A function table shows inputs 1, 2, 3, 4 with outputs 5, 8, 11, 14. Find the function rule.
Find the pattern: Each time the input increases by 1, the output increases by 3. This constant rate of change tells you the multiplier is 3.
Find the starting adjustment: Test the multiplier with the first input: 3 times 1 gives 3, but the output is 5. You need to add 2.
Verify with another input: Check input 4: 3 times 4 plus 2 equals 14, which matches the table.
Answer: The function rule is .
Why It Matters
Finding function rules from tables and patterns is a core skill in middle-school algebra and appears on nearly every standardized math test. Scientists and engineers rely on function rules to model real-world relationships, such as converting temperatures or calculating costs.
Common Mistakes
Mistake: Confusing the function rule with a single input-output pair
Correction: A single pair like (2, 7) does not determine the rule. You need multiple pairs to identify the pattern, because many different rules can produce the same output for one input.