Constructing Inverse Graphs and Real-World Inverses

Lesson 2 of 2: Building and Interpreting f⁻¹

In this lesson:

• Reflect a graph over to construct
• Swap coordinates to build the inverse graph from points
• Interpret values in real-world contexts

What You Will Learn Today

1. Sketch by reflecting over
2. Convert points on to points on
3. Verify using the line
4. Avoid reflecting over the x-axis instead of
5. Interpret in real-world contexts

Deck 1 Recap: What You Already Know

• Table method: scan output column for , read the input →
• Graph method: draw , read x at intersection →
• Invertibility check: no repeated outputs (table) or HLT (graph)

Today: build the full inverse graph from the graph of .

The Reflection Principle: y = x

Reflecting over swaps every point's coordinates:

Swap Coordinates to Build the Inverse

Pick key points from , swap each pair, plot and connect:

Example: Sketching f⁻¹ from Key Points

Points on :

Swapped for :

Plot both, connect smoothly, draw — do the curves mirror each other?

Verify with y = x

• and are symmetric about
• Pick any point on : its reflection over lands on

Quick check: → the point should be on . Is it?

Your Turn: Reflect and Sketch

Given points on :

1. Swap coordinates to find four points on
2. Plot both and on the same axes
3. Draw and verify the reflection

Try before the next slide.

Quick Check: Swap the Coordinates

The point is on .

Where is the corresponding point on ?

Think before you advance.

Watch Out: Wrong Reflection Line

Point reflected over different lines:

Now Apply Inverse to Real-World Context

answers: "what input produces output ?"

• : input gives output
• : output came from input

Worked Example: Hours to Produce Items

items produced in hours.

Find : locate 60 in output → → 4 hours to make 60 items.

Context Example 2: Temperature from Graph

A graph of temperature (°F) at hour during the day is given.

Question: At what hour did the temperature reach 75°F?

Method: draw on the graph, find intersection, read the hour.

Answer: → temperature reached 75°F at hour 10 (10 am).

Your Turn: Interpret f⁻¹ in Context

A function gives the number of calories burned after running miles.

Write a sentence interpreting this result in context.

Think in terms of the quantities: miles and calories.

Quick Check: Write the Interpretation

= distance (km) traveled at time (hours).

Interpret:

What question does this answer? What does 4.5 mean in context?

Advance for the full interpretation.

Practice: Inverse Graph and Context

Problem 1: Points on : . List three points on .

Problem 2: = height (m) of a balloon after minutes. .
Interpret this result in one sentence.

Problem 3: Sketch both and using Problem 1's points. Draw .

Answers: Inverse Graphs and Context Practice

1. Swap pairs:
2. "The balloon reaches 200 m after 8 minutes."
3. Plot for — symmetric with about ✓

Key Takeaways: Inverse Graphs and Context

• Reflect over to get : swap all coordinates
• Verify: and are mirror images across
• Contextual inverse: "what input gives output ?"

Reflect over (swap coordinates), NOT the -axis!

Preview: Restricting the Domain for Inverses

Non-invertible functions can be fixed by restricting the domain:

• on → inverse is
• on → inverse is

Next lesson (HSF.BF.B.4.d): producing invertible functions.