How do you write a function rule in math?

A function rule in math expresses the relationship between inputs and outputs. It's a mathematical way to describe how each input value corresponds to exactly one output value. You can derive function rules from data tables by observing patterns in the relationship between input and output.

How to Write a Function Rule from a Data Table

A function rule is typically written as an equation, where one variable (usually y) depends on another variable (usually x). Here's a step-by-step method for identifying and writing a function rule based on a data table, drawing from the information in the YouTube video "How To Write a Function Rule Given a Data Table | Algebra":

1. Analyze the Data:

   • Look at the provided table and identify the input values (often called 'x') and the corresponding output values (often called 'y').
   • Examine how the x values change and how the y values change. Look for consistent differences or ratios.

2. Identify the Pattern:

   • Try to find a consistent arithmetic operation (addition, subtraction, multiplication, division) or a combination of these that links the input and output values.
   • For example, is each output y obtained by adding a constant value to the input x? Is it a multiple of x? Is there a combination of both involved?
   • The reference video uses the example of a table where the y values are three times the x value, plus two.

3. Write the Rule as an Equation:

   • Once you've identified the pattern, express it as an algebraic equation.
   • Use the variables x to represent input values and y to represent output values.
   • The equation will show how the y value is calculated in terms of the x value.
   • In the example from the video, if the pattern found was "multiply input by 3 and add 2," this translates to the function rule: y = 3x + 2
     • Let's verify with an example using input x = 2: y = 3 * 2 + 2 = 8. This corresponds to what is seen in the example.

Example and Practical Insight

• Suppose your table has these values:

  Input (x)	Output (y)
  0	1
  1	3
  2	5
  3	7

• Analysis: We see that for every increase of 1 in x, the value of y increases by 2. Let us test by seeing if y is a multiple of the input + some value:

  • For input 0, 2 * 0 + 1 = 1 (correct).
  • For input 1, 2 * 1 + 1 = 3 (correct).
  • For input 2, 2 * 2 + 1 = 5 (correct).
  • We can see the y value is always double the x value plus 1.

• Function rule: The function rule will be y = 2x + 1

Key Points

• Function rules must produce only one output for each input.
• A data table is a common method for figuring out a function rule.
• The rule is usually expressed using an algebraic equation.

By using these steps, and based on the information from the YouTube video, you can effectively write function rules from data tables.