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.

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Evaluate composite functions by Khan Academy

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

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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.