Lesson 5-2: Function Tables

Property

A function is a mathematical rule that uses an input number to generate one and only one output number. The relationship between these numbers can be organized in a function table.

To find the output (y) for a given input (x), substitute the value of x into the function's rule and evaluate the expression.

Examples

• For the rule y = 3x - 2, if the input is 2, substitute to get y = 3(2) - 2 = 6 - 2 = 4.
• Complete the table for the rule y = x/4 + 1:

Property

Domain constraints are restrictions on the input values of a function based on real-world limitations or mathematical requirements. Common constraints include non-negative values when representing quantities like time or distance, and whole numbers for countable items.

Examples

• A function C = 50t + 100 represents the cost of renting equipment for t hours. The domain is constrained to t ≥ 0 since negative time doesn't make sense.
• A function P = 15n - 200 represents profit from selling n items. The domain is constrained to non-negative whole numbers since you can't sell a fraction of an item.

Explanation

Real-world functions often have domain constraints that differ from mathematical possibilities. These constraints arise from the physical meaning of the variables involved. For example, time typically cannot be negative, and quantities of objects must be whole numbers. Understanding these limits helps you pick reasonable input values for your function table.

Property

When a function models a real-world scenario, the output y (dependent variable) must be interpreted using the correct units and context associated with the input x (independent variable).

Examples

• A function rule for the total cost of buying n books is C = 12n. If the input is n = 3, the output is C = 36. In context, this means buying 3 books costs $36.
• The remaining battery percentage of a phone after h hours is given by P = 100 - 15h. If the input is h = 2, the output is P = 70. In context, this means after 2 hours of use, the phone has 70% battery remaining.

Explanation

When a function models a real-world situation, the numbers in a function table represent actual quantities with specific units. To interpret an output, you must identify what the dependent variable measures and relate it to the specific input value that produced it. This turns abstract numbers into practical stories!