Reading Inverse Values from Tables and Graphs

Lesson 1 of 2: Finding f⁻¹

In this lesson:

• Find from a table by scanning the output column
• Read inverse values from a graph using horizontal lines
• Check whether a function is invertible

What You Will Learn Today

1. Read from a table
2. Read from a graph — horizontal line method
3. Construct a table of by swapping columns
4. Check whether the inverse exists
5. Interpret : the input that produces output

What Does f⁻¹(b) Actually Mean?

You already know that the inverse swaps inputs and outputs.

• means: input gives output
• means: output came from input

The question: given the output, what was the input?

Finding f⁻¹ from a Table

Scan the output column for the target. Find : locate 8 → row 2 →

Example: Reading Three Inverse Values

• — find 5 in output column, input is 1
• — find 11 in output column, input is 3
• — find 14 in output column, input is 4

Building the Full Inverse Table

Swap columns: each row in becomes in .

Swapped Columns — Side by Side

Verify: and ✓

Every pair in becomes in .

Check Before Building the Inverse Table

Output 7 appears twice → can't choose between 2 and 3 → no inverse.

Try These Three Table Questions

1. Find
2. Find
3. Does exist? How do you know?

Try each before the next slide.

Same Question, Different Representation: Graphs

The same question applies to graphs: given output , what input gives ?

Strategy: draw a horizontal line at height and find where it crosses the graph.

The x-coordinate of that intersection is .

Horizontal Line Method: Find f⁻¹ from a Graph

Draw , find intersection at →

Reading Multiple Inverse Values from a Graph

Each row: draw , read x at intersection. These points trace the graph of .

When the Inverse Doesn't Exist — Graphs

Horizontal line hits the graph twice → two possible inputs for one output.

The inverse is not well-defined on the full domain.

Example: Parabola Fails Horizontal Line Test

For on all real numbers:

• and
• → could be or

The inverse can't choose. The full parabola fails the HLT.

In Deck 2, you'll learn how to fix this by restricting the domain.

Your Turn: Read f⁻¹ from a Graph

A graph of f is shown. Use horizontal lines to answer:

1. Find
2. Find
3. For what value of does not exist?

Annotate each horizontal line on the graph before checking.

Quick Check: Does the Graph Have an Inverse?

Graph A: A strictly increasing line
Graph B: A parabola opening upward
Graph C: A horizontal line

1. Which graphs pass the horizontal line test?
2. For which graphs does exist on the full domain?

Think before the next slide.

Practice: Tables, Graphs, and Invertibility

1. Find and .
2. Line hits graph of at . Find .
3. Outputs: {4, 4, 9, 16} — inverse exists?

Answers: Check Your Practice Work

1. and — scan output column, read corresponding input
2. — line hits graph at
3. Output 4 appears twice → no inverse exists

What You Learned: Two Reading Methods

• Table: find in the OUTPUT column → read the input
• Graph: draw → find intersection → read x-coordinate
• Check: repeated outputs or HLT failure → no inverse

: search outputs, not inputs!

Coming Up Next: Building the Inverse Graph

Deck 2 — constructing and interpreting :

• Reflect the graph of over the line
• Point on maps to point on
• Apply inverse reading to real-world contexts