Functions
A rule assigning one output to each allowed input.
Use a function as an input–output rule
Substitute values, follow the order of operations and check the permitted inputs.
By the end: Calculate a function output and distinguish the rule from one observed pair.
Try the lab first · Read the explanation
Starting recap and optional refresher
Multiply before adding. A function assigns one output to each allowed input.
Review substitution and arithmetic, then return hereTry a concrete example
A workshop charges 3 credits to start and 2 credits per item: f(x) = 2x + 3. For four items, the cost is 11 credits.
Make a prediction
Before changing the rule, predict one output and write down what you expect to change. Check the table after your intervention. This prediction is practice, not an assessment attempt.
Change one thing and compare
Keep slope 2 and intercept 3. Predict the outputs at 0, 1 and 4 before moving the input. Change the intercept only and compare every table row.
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A rule you can use repeatedly
A function assigns one output to each permitted input. We write f(x) to mean “the output of the rule f at input x.” It is not f multiplied by x. A function may be given by a formula, a table or a piecewise instruction.
For the workshop rule f(x) = 2x + 3, x counts items and f(x) is the total in credits. The multiplier has units credits per item. The added three has units credits. Adding quantities with unrelated units would not make a coherent price rule.
Substitute, multiply, add
At x = 4, replace x with 4: f(4) = 2 × 4 + 3 = 11 credits. At x = 0, the output is 3. This starting fee remains even when there are no items; it is not an extra charge per item.
Try a second input to distinguish a rule from a memorized answer. At x = 1, the output is 5, while at x = 2 it is 7. Increasing the input by one changes the output by two for this rule.
The allowed inputs matter
A formula can return a number outside a useful real-world range. Negative item counts are not valid orders. A discount might change the rule beyond ten items, and a capacity limit might disallow more than twenty. These conditions belong to the model’s domain: the set of inputs for which you intend to use it.
Changing a slider changes the authored model. It does not fit a model to evidence. In AI, a similar expression may use a learned weight and bias; here you choose the numbers directly so their roles are visible.
Try a new case
Check your understanding
For f(x) = 3x + 2, what is f(4)?
Take the idea with you
Calculate a function output and distinguish the rule from one observed pair.
Completion records that you finished the lesson. It does not certify independent proficiency.
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Original LearnAI mathematics lesson. All experiments run locally on the CPU. No refresher practice or AI achievement is changed by completing this lesson.
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Slope
Slope can review Functions.
Read input and output values before comparing how much output changes per input unit. Distinguish an interval rate from a local derivative.
Scope and source review
Graph reading and rates in the five introductory mathematics lessons.
Source-audited 2026-10-31. Independent subject-expert and learner review remain pending.
Linear and affine functions
Linear and affine functions extends Functions.
Straight-line rules provide a concrete family of functions with constant rate and an optional starting value; not every function is linear or affine.
Scope and source review
Affine y = mx + b rules in the introductory function lessons; strictly linear rules have zero intercept.
Source-audited 2026-10-31. Independent subject-expert and learner review remain pending.
11 concepts in this cross-subject pilot, not a complete curriculum graph. The existing AI lesson map and progress records remain separate and available below or through AI discovery.