Mastery Check Solutions

Show What You Know

Two groups of students (Red and Blue) want to determine if the functions f (x) and g(x) are inverses of one another. Both groups decided to use $x = 7$ and stated that f (x) and g(x) were inverses.

The teacher told both groups to look more closely at the problem because f(x) and g(x) are not inverses.

The Red group decided to switch the domain and range. Show the work that the Red group would have completed.

$\begin{array}{rcl} y & = & \frac{5}{3} (x - 4) + 2 \\ x & = & \frac{5}{3} (y - 4) + 2 \\ \frac{3}{5} (x - 2) & = & (\frac{5}{3} (y - 4)) (\frac{3}{5}) \\ \frac{3}{5} (x - 2) & = & y - 4 \\ y & = & \frac{3}{5} (x - 2) + 4 \end{array}$

Note

Q: How is your inverse of $f (x)$ different from $g (x)$ ? Use a, h, k to help explain.

A: For $f^{- 1} (x) : a = \frac{3}{5}, h = 2, k = 4$

For $g (x) : a = \frac{3}{5}, h = k = 7$

If $f (h) = k$ , then $f^{- 1} (k) = h$

The Blue group decided to use a calculator when $x = - 2$ and $x = 10 .$ Show the work that the Blue group would have completed.

$f (- 2) = \frac{5}{3} (- 2 - 4) + 2 f (- 2) = - 8 g (- 8) = \frac{3}{5} \sqrt{- 8 - 7} + 7 g (- 8) = undefined$

$f (10) = \frac{5}{3} (10 - 4) + 2 f (10) = 12 g (12) = \frac{3}{5} \sqrt{12 - 7} + 7 g (12) = 8.34$

Note

Use a calculator to evaluate the function at the given values. Write the problem before using a calculator so you know the values to input.

Using parts A and B, explain why $f (x)$ and $g (x)$ are not inverses of one another.

Sample: When the Red group switched the domain and range of $f (x)$ , the result was not g(x). This is because g(x) has a square root symbol and $f (x)$ does not.

When the Blue group used other values of x to verify inverses, they found $f (- 2) = - 8$ and $g (- 8) = undefined$ . Since the ordered pairs were not $(a, b)$ and $(b, a)$ , the given functions are not inverses.

Note

Q: Explain why $(7, 7)$ makes the given functions appear to be inverses.
A: The ordered pair $(7, 7)$ is the point that both functions share. But the inverse of a linear equation cannot be a square root equation and vice versa.

Say What You Know

In your own words, talk about what you have learned using the objectives for this lesson and your work on this page.

Note

Restate the objectives of the lesson in your own words. If you are unable to restate the lesson objectives, go back and reread the objectives and then explain them.

Determine if functions are inverses of one another algebraically.
Find the inverse of a simple function algebraically.