Mr. Integral

THE BIG IDEA

Functions Review

What does a function tell us—and what can it hide?

A function is an input–output rule with one output per allowed input. Build its vocabulary first, then study graph transformations, composition and reversibility as different ways of using that rule.

Begin the first lesson →See the lesson route ↓

27 lessons · No deadlines or locked lessons

SEE THE CONNECTIONS

How the ideas fit together.

A map of the main ideas. Follow a node to its lesson.

  1. Start with01Inputs & outputs
  2. plotting the rule reveals02Graph behavior
  3. chaining rules creates03Composition
  4. undoing a rule requires04Inverse functions

YOUR LEARNING ROUTE

One idea at a time.

Read the explanation, use the visualization, then try the check before opening its reasoning.

Keep the unit cheat sheet handy →
Read and evaluate functions7 lessons
  1. 01Defining a FunctionIdentify a function by the uniqueness of each input’s output.
  2. 02Evaluating a FunctionEvaluate a function by substituting its input consistently.
  3. 03Domain, Codomain and RangeDistinguish inputs, declared outputs and attained outputs.
  4. 04What Makes a Rule a FunctionUse the vertical-line test to check a relation.
  5. 05Properties of the RangeRelate the range of a function to its codomain.
  6. 06Reading Domain and Range off a GraphRead domain and range from the horizontal and vertical extent of a graph.
  7. 07Domain and Range of a Rational FunctionFind restrictions on a rational function’s inputs.
Recognize function families3 lessons
  1. 08Linear FunctionsInterpret the slope and intercept of a linear graph.
  2. 09Piecewise FunctionsChoose the correct branch when evaluating a piecewise function.
  3. 10Writing an Absolute Value PiecewiseRewrite absolute values using the sign of the inner expression.
Read symmetry and transformations6 lessons
  1. 11Even FunctionsRecognize even functions from algebra and symmetry.
  2. 12Odd FunctionsRecognize odd functions from algebra and origin symmetry.
  3. 13Arithmetic on Even and Odd FunctionsPredict parity when combining even and odd functions.
  4. 14Symmetry of a GraphRead reflected points from a graph’s symmetry.
  5. 15Translating a GraphTranslate graphs by tracking how an input is replaced.
  6. 16Stretching and Compressing a GraphDistinguish vertical scaling from changes to the input.
Combine and invert functions9 lessons
  1. 17Defining CompositionEvaluate a composition in the correct order.
  2. 18Properties of CompositionCheck the order and domain of a composition.
  3. 19One-to-One FunctionsTest whether distinct inputs remain distinguishable by their outputs.
  4. 20Functions That Are Not OntoIdentify codomain values that a function does not reach.
  5. 21Onto FunctionsDetermine whether every declared output is attained.
  6. 22Why One-to-One and Onto MatterExplain how bijections make reversible input–output rules possible.
  7. 23Finding the Inverse of a FunctionFind an inverse by solving for the original input.
  8. 24A Shortcut for the InverseUse an inverse shortcut while retaining its nonzero-slope condition.
  9. 25A Function and Its Inverse on the GraphRelate inverse graphs by reflection across y=x.
Describe the behavior of a graph2 lessons
  1. 26Increasing and Decreasing IntervalsIdentify increasing and decreasing intervals from output comparisons.
  2. 27Local and Absolute Maxima and MinimaDistinguish local behavior from extrema over an entire domain.