Unit 1 of 10

Functions, Relations, and Graphs

  • 17 lessons
  • Parts 1 to 34

A function is a rule that gives exactly one output for each input. Nearly everything in precalculus and calculus is a statement about functions, so this unit builds the language with care, from sets and intervals on the number line to domains, ranges and graphs.

You will read graphs for intercepts, rising and falling intervals and turning points, measure average rates of change with secant lines, and work with piecewise, absolute value, even and odd functions. The unit ends with shifting and stretching graphs, combining and composing functions, and undoing a function with its inverse.

Every lesson is a pair of videos. The theory part explains the idea and the practice part works through problems. Each lesson page has the definitions and an example for every key term.

Sets and Functions

  1. Sets, Intervals, and Inequalities

    How to describe sets of real numbers by listing, by set-builder notation and by intervals, combine them with union and intersection, and solve inequalities without losing track of their direction.

    Theory part 1Practice part 2

  2. Relations, Functions, and the Vertical Line Test

    What relations and functions are as sets of ordered pairs, how inputs, outputs and the domain fit together, and how the vertical line test spots an input with two outputs.

    Theory part 3Practice part 4

  3. Function Notation and Evaluation

    How to read f(x) as the output of a function at the input x, evaluate a function from a formula, a table or a graph, and substitute whole expressions such as a + h correctly.

    Theory part 5Practice part 6

  4. Natural and Specified Domains

    How to find the inputs a formula accepts by ruling out zero denominators and negative numbers under even roots, and why a domain stays the same after you cancel a common factor.

    Theory part 7Practice part 8

  5. Range and Codomain

    How the codomain of a function differs from its range, how to find the range by solving for the input or reading the graph, and how bounds and a smaller domain shape it.

    Theory part 9Practice part 10

Rates and Graph Behavior

  1. Linear Functions and Constant Rate of Change

    What makes a function linear, how its slope measures a constant rate of change with units, how to write a line in slope-intercept and point-slope form, and why a vertical line is not a function.

    Theory part 11Practice part 12

  2. Reading Graphs and Function Behavior

    How to read intercepts, rising and falling intervals, signs and highest and lowest points from a graph, why the scale matters, and when a value read from a picture needs an exact check.

    Theory part 13Practice part 14

  3. Average Rate of Change and Secant Lines

    How the average rate of change measures output change per unit of input change over an interval, why it equals the slope of a secant line, and how to read its sign and units.

    Theory part 15Practice part 16

Special Kinds of Functions

  1. Piecewise-Defined Functions

    How one function can use different formulas on different parts of its domain, how to pick the right formula for an input, and how to read the open dots, closed dots and jumps where the formulas change.

    Theory part 17Practice part 18

  2. Absolute Value Functions, Equations, and Inequalities

    How absolute value measures distance on the number line, why its graph is a V, and how to solve absolute value equations and inequalities, including the cases that have no solutions or false ones.

    Theory part 19Practice part 20

  3. Even, Odd, and Symmetry

    How to tell from a formula or a graph whether a function is even, odd or neither, why the domain has to be symmetric, and how mirror symmetry and half-turn symmetry of a graph match these algebraic tests.

    Theory part 21Practice part 22

Transformations of Graphs

  1. Shifts and Reflections of Graphs

    How adding a constant inside or outside a function moves its graph left, right, up or down, how a minus sign reflects it across an axis, and how to follow points through several such moves.

    Theory part 23Practice part 24

  2. Stretches and Compressions of Graphs

    How multiplying the outputs or the inputs of a function stretches or compresses its graph, why a multiplier on the input acts by its reciprocal, and how to track points when stretches and shifts are combined.

    Theory part 25Practice part 26

Combining and Inverting Functions

  1. Algebra of Functions

    How to add, subtract, multiply and divide functions one input at a time, how to find the domain of the result, and why a restriction stays even after the formula simplifies.

    Theory part 27Practice part 28

  2. Function Composition

    How to feed the output of one function into another, why the order matters, how to find the domain of a composite, and how to split a formula into an inner and an outer function.

    Theory part 29Practice part 30

  3. One-to-One Functions and the Horizontal Line Test

    What it means for a function to be one-to-one, how to prove it with algebra or disprove it with one repeated output, how the horizontal line test reads it from a graph, and why strictly increasing functions pass.

    Theory part 31Practice part 32

  4. Inverse Functions and Domain Restriction

    What an inverse function is, why its domain is the range of the original, how to find its formula and graph, and how to restrict a function that is not one-to-one so it can be inverted.

    Theory part 33Practice part 34