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    • ▾Precalc
      • ▸Conic sections
        • •Intro to conic sections
        • •Center and radii of an ellipse
        • •Foci of an ellipse
        • •Intro to hyperbolas
        • •Foci of a hyperbola
        • •Maximizing the product of two integers, given their integer sum
        • •Hyperbolas not centered at the origin
        • •Standard equation of a circle
      • ▾Functions and relations
        • •Composition of functions
        • •Operations with functions
        • •Graphing the sum of trig functions from their graphs
        • •Finding and verifying the inverse of a nonlinear function
        • •Graphically finding the increasing, decreasing, and constant intervals
        • •Vertical and horizontal line tests
        • •Restrict the domain then find the inverse
        • •Sketching the inverse of a function from its graph
      • ▸Characteristics of functions
        • •Finding extrema graphically
        • •End behavior
      • ▸Rational functions
        • •Partial fraction decomposition
        • •Simplifying the difference quotient
        • •Pick's theorem
      • ▸Sigma notation and the binomial theorem
        • •Converting sums between sigma notation and expanded form
        • •Properties of sigma notation
        • •Using the binomial theorem
      • ▸Series
        • •Geometric series
        • •Geometric series with summation notation
        • •Arithmetic series with summation notation
      • ▸Logic
        • •De Morgan's laws for propositional logic
     › Precalc › Functions and relations

    Composition of functions

    First, students will be introduced to the idea of composite functions.

    Intro to composing functions by Khan Academy

    Next, students will learn how to evaluate composite functions for numeric inputs, by substituting, then evaluating.

    Watch these Khan Academy videos:

    • Evaluating composite functions
    • Evaluating composite functions (advanced)

    Evaluate composite functions by Khan Academy

    After that, students will learn how to evaluate composite functions using graphs and tables.

    Watch these Khan Academy videos:

    • Evaluating composite functions: using tables
    • Evaluating composite functions: using graphs

    Evaluate composite functions: graphs & tables by Khan Academy

    Next, students will learn how to find composite functions algebraically.

    Finding composite functions by Khan Academy

    Find composite functions by Khan Academy

    After that, ask your students to prove that the composition of two linear functions is always linear. Don't ask your students to prove the composition of quadratic functions is quartic (has degree \(4\)), because the algebra is quite messy. That's because they would need to square a quadratic. Next, teach your students what total and partial functions are, and thus why the composition of two total functions is always total.

    Conclude by giving your students these challenges:

    • Maze 100 by NRICH
    • Picturing the World by NRICH
    • Mathsjam Jars by NRICH
    • Removing Square Tiles

    2021 Math Kangaroo Levels 11-12 Problem #21: The figure shows the graph of a function \(f,\) defined over the interval \([-5, 5].\) How many distinct solutions does the equation \(f(f(x)) = 0\) have?

    Here's the solution.

    Additional lessons and practice problems