Evaluating piecewise functions

Students will learn how to evaluate unary piecewise functions, including the absolute value function, and piecewise linear functions. They'll evaluate using either the function's definition or its graph. The second problem in this video is a word problem requiring students to evaluate a piecewise function from its graph.

Watch these Khan Academy videos:

Evaluate piecewise functions by Khan Academy

Next, students will learn how to model piecewise functions from word problems (video).

After that, students will learn the definitions of $\text{min}(x,y)$ and $\text{max}(x, y),$ and how to evaluate these functions for various inputs. For example, $\text{max}(3, 2) = \mathop{?}$

Next, give your students these challenges:

Express abs in terms of max. The answer is

$$\lvert x \rvert = \max(x, -x)$$

Express $\max$ in terms of $\text{abs}.$ Then use your answer to express $\max(x, 0)$ and $\min(x, 0)$ in terms of $\text{abs}.$ The answer is

$$\begin{align} & \max(x, y) = \dfrac{x + y + \lvert x - y \rvert}{2} \\[1em] & \max(x, 0) = \dfrac{x + \lvert x \rvert}{2} \\[1em] & \min(x, 0) = \dfrac{x - \lvert x \rvert}{2} \end{align}$$

The challenge is a combination of this and this.

Mentally determine what fraction is shaded.

Answer: $(3/16)\pi$

Conclude by leading this investigation:

The Broken Sword (addition, division, mini-mathematical universe)
by MathPickle

Lessons and practice problems