Finding Inverses of Restricted Functions

Lesson 2 of 2: Swap, Solve, and Explore

In this lesson:

• Apply swap-and-solve to a restricted function
• State domain and range of the resulting inverse
• Discover how different restrictions produce different inverses

What You Will Learn Today

1. Apply the swap-and-solve method to a restricted function
2. State the domain and range of using the flip rule
3. Explain why you choose the positive or negative root when solving
4. Find both inverses from two different domain restrictions
5. Explain why inverse function conventions exist

Deck 1 Recap: Restriction Works

• Non-invertible functions fail HLT — two inputs map to same output
• Restriction: keep one branch, cut at the turning/symmetry point
• The formula is unchanged; only allowed inputs change

Now: invert the restricted function using swap-and-solve.

Domain and Range Flip Rule

• Domain of restricted → Range of
• Range of restricted → Domain of

For , : both domain and range are , so both flip to .

Worked Example: Finding the Restricted Inverse

, . Range of : .

Step-by-Step Swap and Solve Method

, :

Step 1: ; Step 2: swap:

Step 3: solve: (positive root — restriction)

Why You Must Take the Positive Root

For , :

• Range of = domain of =
• must output non-negatives → positive root only

The restriction — not algebra — determines the root choice.

Guided Practice: Find the Restricted Inverse

Find for , .

1. State the range of (domain of )
2. Apply swap-and-solve
3. State domain and range of

Try all three steps before advancing.

Quick Check: Domain of f⁻¹

restricted to has range .

What is the domain of ?

Think — then advance.

Does the Range Change When You Restrict?

Bounded restrictions shrink the range. Unbounded half-domains may not.

Transition: Two Restrictions, Two Inverses

You've computed for one restriction of .

What if you choose the OTHER restriction?

The formula changes. You get a different inverse function.

This is the core insight of this standard.

Different Restrictions Produce Different Inverses

: two valid inverses from two restrictions.

Verify Both Branch Inverses Work

:

and ✓

:

and ✓

Why Conventions Pick One Valid Inverse

For : two valid inverses, but conventions choose one.

• Convention: right half,
• is defined non-negative for exactly this reason

Math allows multiple; convention picks one standard.

Guided Practice: Find Both Inverses

Find both inverses of (using both restrictions):

1. Restriction : find and state domain/range
2. Restriction : find and state domain/range

Show your work for both before advancing.

Practice: Restrict, Invert, and State

Problem 1: ,
Find . State domain and range of .

Problem 2: ,
Find . State domain and range of .

Answers: Restrict, Invert, and State

P1: , domain , range

P2: , domain , range

What You Learned: Restricted Inverses

• Swap-and-solve after restriction; justify which root you take
• Domain/range of flip from restricted
• Different restrictions → different valid inverses

The restriction — not algebra — determines the root!

What's Next: Logarithms as Inverses

HSF.BF.B.5: inverses of exponential functions

• is one-to-one on all reals → inverse exists without restriction
• Logarithms are inverses of exponentials — domain restriction not needed!

You're ready for the next step in building inverse functions.