Domain Restriction for Invertible Functions

Lesson 1 of 2: Why and How to Restrict

In this lesson:

• Identify functions that fail the horizontal line test
• Understand why restriction creates an invertible function
• Choose the right domain restriction

What You Will Learn Today

1. Explain why non-one-to-one functions need domain restriction
2. Identify symmetry or turning points that cause the HLT to fail
3. Restrict the domain to produce a one-to-one function
4. Choose restrictions that preserve useful portions of the function
5. Recognize that restriction changes allowed inputs, not the formula

Can Every Function Be Inverted?

You know from HSF.BF.B.4 that the inverse swaps inputs and outputs.

But what if two inputs map to the same output?

• and
• Then → can't decide between 3 and

The inverse is not well-defined — it can't choose.

The Horizontal Line Test Failure

Any horizontal line above the vertex hits the parabola twice → not one-to-one.

Other Functions That Fail HLT

The symmetry or periodicity is the culprit in each case.

Which Functions Pass the HLT?

1. — positive-slope line
2. — upward parabola
3. , — right half of parabola
4. — S-shaped cubic

Which pass? Which fail? Try before advancing.

The Fix: Restrict the Domain

Restricting the domain means limiting the allowed inputs so each output comes from only one input.

• on all reals → HLT fails
• on → HLT passes

The rule is unchanged. Only the set of inputs changes.

Restriction Makes the Inverse Unambiguous

With restricted to :

• is excluded; is the only input giving 9
• — no ambiguity

Transition: Choosing the Right Restriction

You know restriction works. But has many possible restrictions.

Question: how do you choose WHICH interval to restrict to?

Answer: find the symmetry point or turning point and cut there.

Choosing the Restriction for x²

• : right half, inverse is ← standard convention
• : left half, inverse is
• : partial right half, inverse is on

All three are mathematically valid. Convention picks .

Restrict at the Turning Point, Not x=0

, vertex at :

• ✓ or → one-to-one
• ✗ → vertex inside domain → fails HLT

Verifying Restrictions for a Shifted Parabola

, vertex at :

• : one-to-one ✓ →
• : one-to-one ✓ →
• : vertex inside domain → fails HLT ✗

Why Math Conventions Pick One Restriction

Mathematics allows many valid restrictions. Conventions choose the most natural:

• : prefer → inverse is
• : prefer → inverse is

Guided Practice: Choose the Restriction

For each function, locate the turning/symmetry point and choose a restriction:

1. — where is the vertex?
2. — where is the "corner" (symmetry point)?

State the restriction and verify it produces a one-to-one function.

Quick Check: Why Not x ≥ 0 Here?

Why is NOT the right restriction?

Explain in one sentence before advancing.

Practice: Find Vertex and Choose Restriction

1. — where is the vertex?
2. — where is the corner?
3. — where is the vertex?

State the symmetry point and your restriction for each.

Answers: Vertex and Restriction Practice

1. : vertex → or
2. : corner → or
3. : vertex → or

What You Learned: Restriction Strategy

• Functions that fail HLT cannot be inverted on their full domain
• Restriction: keep one side of the symmetry/turning point
• The formula stays the same — only allowed inputs change

Always locate the turning point first — don't default to !

Coming Up Next: Finding the Inverse

Deck 2 — finding of the restricted function:

• Apply the swap-and-solve method after restriction
• Determine domain and range of
• Explore how different restrictions give different inverses