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Read Inverses from
Graphs or Tables

Learn to find inverse function values by reading graphs and tables: swap inputs and outputs to unlock \( f^{-1}(x) \).

Algebra 2

What Is an Inverse Function?

An inverse function \( f^{-1} \) reverses the action of \( f \). If \( f \) sends an input \( a \) to an output \( b \), then \( f^{-1} \) sends \( b \) back to \( a \).

\( f \)
Takes \( x \) and produces \( y \)
\( f^{-1} \)
Takes \( y \) and returns \( x \)

Visualizing the Swap

\( f \)
 input → output swap
\( f^{-1} \)
 output → input
\( f(2) = 5 \)
\( f^{-1}(5) = 2 \)

If your function table shows \( f(2) = 5 \), then the inverse table must show \( f^{-1}(5) = 2 \). Every ordered pair \((a, b)\) on \( f \) becomes \((b, a)\) on \( f^{-1} \).

When Does an Inverse Exist?

A function has an inverse if and only if it is one-to-one: each output corresponds to exactly one input. Graphically, this means it passes the horizontal line test.

✓ One-to-One
No two \( x \)-values share the same \( y \)-value. Every horizontal line crosses the graph at most once.
✕ Not One-to-One
Some \( y \)-value is hit by more than one \( x \)-value. A horizontal line crosses the graph twice or more.

The Core Rule

x y
\( y = x \)
\( f \)
\( f^{-1} \)
 (a, b) (b, a)

The graph of \( f^{-1} \) is the reflection of \( f \) across the line \( y = x \). Every point flips its coordinates.

Reflection Across \( y = x \)

x y 1 2 3 4 -1 -2 1 2 3 -1 -2
\( y = x \)
 (1, 3) (3, 1) (-1, -2) (-2, -1)

Table vs. Graph

From a Table
Find the row where the output column shows the value you need. The corresponding input is your answer.
From a Graph
Start at the given \( y \)-value on the vertical axis, move horizontally to the curve, then drop down to read the \( x \)-value.

How to Apply the Swap

1
Identify the question. You need \( f^{-1}(k) \) for some value \( k \).
2
Locate \( k \) in the output. In a table: scan the \( f(x) \) column for \( k \). On a graph: find \( y = k \) on the vertical axis.
3
Read the corresponding input. In a table: the \( x \)-value in that row is your answer. On a graph: trace horizontally to the curve, then vertically to the \( x \)-axis.
4
Write the result: \( f^{-1}(k) = \) that input value.

Example 1: Reading a Table

\( x \) -2 0 1 3 5
\( f(x) \) 6 4 2 -1 -3

Find \( f^{-1}(2) \).

Example 2: Reading a Graph

1 2 3 4 5 -1 -2 -3 1 2 3 -1 -2 -3
\( f \)
\( y = 1 \)
\( x = 1 \)

Example 3: Building an Inverse Table

Given this table for \( g \), construct the table for \( g^{-1} \):

\( x \) 0 1 2 3 4
\( g(x) \) 5 3 7 1 9
\( x \) 5 3 7 1 9
\( g^{-1}(x) \) 0 1 2 3 4

Example 4: Multiple Readings

1 2 3 4 5 -1 -2 -3 1 2 3 4 5 -1
\( h \)
 (-2, 0) (0, 1) (1, 3) (3, 4) (5, 5)

Domain and Range Connection

When you swap inputs and outputs, the domain and range also swap:

Domain of \( f \)
Becomes the range of \( f^{-1} \)
Range of \( f \)
Becomes the domain of \( f^{-1} \)

The Composition Check

A powerful way to verify your inverse reading:

✓
Read \( f^{-1}(k) = a \).
✓
Verify: Go back to the graph or table and confirm \( f(a) = k \).
✓
If both match, the inverse reading is correct.

Common Mistakes

✕ Looking in the Wrong Column
For \( f^{-1}(k) \), students search for \( k \) in the \( x \)-row instead of the \( f(x) \)-row. Remember: you are looking for \( k \) in the output.
✕ Reading Axes Backwards
On a graph, students start at the \( x \)-axis and go up for \( f^{-1} \). Instead, start at the \( y \)-axis and go horizontally to the curve.
✕ Confusing \( f^{-1}(x) \) with \( \frac{1}{f(x)} \)
The notation \( f^{-1} \) means inverse function, not the reciprocal. \( f^{-1}(3) \neq \frac{1}{f(3)} \).

Summary at a Glance

From a Table From a Graph
Goal Find \( x \) where \( f(x) = k \) Find \( x \) where \( y = k \) meets the curve
Locate \( k \) Scan the output row Start at \( y = k \) on the y-axis
Read the answer The matching input value Drop down to the x-axis
Verify Check \( f(a) = k \) in the table Confirm \( (a, k) \) lies on the curve

Question 1 of 5

The table below shows values of \( f \). What is \( f^{-1}(7) \)?

\( x \) 1 2 3 4 5
\( f(x) \) 3 7 11 15 19

Question 2 of 5

If the point \( (4, -2) \) is on the graph of \( f \), which point is on the graph of \( f^{-1} \)?

Question 3 of 5

The graph of \( g \) passes through the points \( (-1, 0) \), \( (0, 2) \), \( (2, 6) \), and \( (3, 8) \). What is \( g^{-1}(6) \)?

Question 4 of 5

Given the table for \( p \), find \( p^{-1}(p(5)) \).

\( x \) 2 5 8 10
\( p(x) \) -3 4 9 12

Question 5 of 5

A one-to-one function \( f \) has domain \( [-3, 5] \) and range \( [1, 9] \). What is the domain of \( f^{-1} \)?