Functions

This course is a part of Algebra (all content), a 20-course Topic series from Khan Academy.

This topic covers: - Evaluating functions - Domain & range of functions - Graphical features of functions - Average rate of change of functions - Function combination and composition - Function transformations (shift, reflect, stretch) - Piecewise functions - Inverse functions - Two-variable functions

This course contains 30 segments:

Evaluating functions

Learn how to find the value of a function for a given input value.

Inputs and outputs of a function

Extend your understanding of the relationship between the inputs of a function and the outputs of that function.

Functions and equations

Understand the subtle differences and similarities between functions and equations. In this exercise, we will see how an equation can be turned into a function.

Interpreting function notation

Solve some word problems by interpreting expressions of modeling functions.

Introduction to the domain and range of a function

Learn what the domain and the range of a function are. Practice finding the domain and the range of a function given its graph.

Determining the domain of a function

Determine the domains of functions according to various considerations.

Recognizing functions

Learn how to distinguish between two-variable relationships that are functions and two-variable relationships that aren't functions.

Piecewise functions

Piecewise functions are functions that have different assignment rules for different numbers. For example, a function can return x² for any negative x-value and return x³ for any non-negative x-value.

Maximum and minimum points

Learn about maximum and minimum points of functions.

Intervals where a function is positive, negative, increasing, or decreasing

Learn about features of functions that have distinct graphical representations: intervals where the function is always positive or always negative, and intervals where the function is always increasing or always decreasing.

Interpreting features of graphs

Interpret the graphs of functions in terms of the contexts that are modeled by the functions.

Average rate of change

Learn what's the average rate of change of a function and how to find it over given intervals.

Average rate of change word problems

Solve word problems that concern the average rate of change of a functional relationship.

Combining functions

Learn how to take two basic functions and combine them into a new function which is their sum, difference, product, or quotient.

Composing functions (Algebra 2 level)

Learn what function composition is, how to to work with composite functions, and how to find the formulas of the composition of two functions.

Shifting functions

Learn how to transform functions so their graph is shifted horizontally or vertically. Learn how to find the equation of a function that is a shift of another function.

Stretching functions

Learn how to transform functions so their graph is stretched horizontally or vertically. Learn how to find the equation of a function that is a stretch of another function.

Modeling situations by combining and composing functions (Algebra 2 level)

Make use of knowledge in combining and composing functions, by modeling complex real-world situations.

Introduction to inverses of functions (Algebra 2 level)

Learn what inverses of functions are, and gain some experience with them by finding the outputs of the inverse of a function given that function.

Finding inverse functions (Algebra 2 level)

Learn how to find the inverses of polynomials functions, radical functions, and rational functions.

Verifying that functions are inverses (Algebra 2 level)

Learn how to check whether a pair of functions are the inverses of one another.

Determining whether a function is invertible (Algebra 2 level)

Learn about the conditions for a function to be invertible, and learn how to manipulate functions to make them invertible.

Determining the domain of advanced functions (Algebra 2 level)

Learn how to determine the domain of piecewise functions with different function rules.

Determining the range of a function (Algebra 2 level)

Learn how to find the range of a function, which is the set of all of the function's possible outputs.

Graphing nonlinear piecewise functions (Algebra 2 level)

Learn how to graph piecewise functions whose assignment rules are not linear.

Interpreting the symmetry of algebraic models (Algebra 2 level)

Learn how to interpret graphs of modeling functions in terms of their context by considering their symmetry.

Interpreting the end behavior of algebraic models (Algebra 2 level)

Learn how to interpret graphs of modeling functions in terms of their context by considering their end behavior.

Interpreting the periodicity of algebraic models (Algebra 2 level)

Learn how to interpret graphs of modeling functions in terms of their context by considering their periodicity.

Comparing features of functions (Algebra 2 level)

Learn how to compare various features between two functions, each represented in a different way.

Two-variable functions

Learn about functions that take *two* inputs in order to output a single value.