Lesson 9 Piece Out Practice Understanding

Learning Focus

Write and graph piecewise functions efficiently and accurately.

Write and graph absolute value functions in piecewise and absolute value form.

Find inverses and graph inverse functions efficiently and accurately.

How do piecewise functions, absolute value functions, and inverse functions connect?

Open Up the Math: Launch, Explore, Discuss

In this unit, we started with piecewise functions, connected to absolute value functions, and found inverse functions. In this lesson, we are going to put all this work together by starting with a single function and building the other functions and their representations from the initial function. Think flexibly, and don’t forget to check your work!

1.

Start with $f (x) = - (x + 3) - 1$

A. Find $f^{- 1} (x)$ :

B. Graph the piecewise function:

$h (x) = {\begin{cases} - (x + 3) - 1, & - 5 \leq x < 2 \\ 2 (x - 2) - 6, & 2 \leq x < 5 \\ - \frac{1}{2} (x - 5), & 5 \leq x \leq 10 \end{cases}$

C. Write an absolute value function, $g (x),$ in piecewise form with $f (x)$ as one side of the function:

Write $g (x)$ in absolute value form:

2.

Start with $f (x) = 2 \| x - 3 \| + 5$	A. Graph $f (x)$ :
B. Write the equation of the piecewise function for $f (x)$ .	C. Find the inverse of $g (x) = 2 (x - 3) - 5$ :

3.

Start with $f (x) = - {(x + 2)}^{2} + 3$ for $x \geq - 2$	A. Find $f^{- 1} (x)$ :
B. Graph $g (x) = {\begin{cases} 3, & - 6 \leq x < - 2 \\ - {(x + 2)}^{2} + 3, & - 2 \leq x < 2 \\ - 5, & 2 \leq x < 6 \end{cases}$

4.

Start with $f (x) = {\begin{cases} - 2 x + 1, & - 3 \leq x < 0 \\ - \frac{1}{2} x + 1, & 0 \leq x \leq 3 \end{cases}$	A. Graph $f (x)$ and $f^{- 1} (x)$ :
B. Graph $g (x) = - 2 \| x \| + 1$	C. Write $g (x)$ in piecewise form:

Ready for More?

Let $f (x) = {\begin{cases} - 2 x + 1, - & 3 \leq x < 0 \\ - \frac{1}{2} x + 1, & 0 \leq x \leq 3 \end{cases}$ .

Write the equation of $f^{- 1} (x)$ , the function you graphed in problem 4a.

Takeaways

Helpful suggestions to be more efficient, accurate, and flexible with piecewise, absolute value, or inverse functions:

Lesson Summary

In this lesson, we worked on becoming more fluent with piecewise functions, absolute value functions, and inverse functions. We learned to be more efficient in our work and to refine the details so that it is entirely correct.

Retrieval

Use properties of exponents to write an equivalent expression. Write your answer in exponential form with positive exponents.

1.

Use properties of exponents to write an equivalent expression. Write your answer in exponential form with positive exponents.

$\sqrt[5]{x^{2}} \cdot \sqrt[3]{x}$

2.

Given: $f (x) = 5 x - 7$ and $g (x) = \frac{(x + 7)}{5}$

Calculate $f (g (x))$ and $g (f (x))$ for the pair of functions. Are $f (x)$ and $g (x)$ inverses?