📌 What is an Inverse?
An inverse relation "undoes" a given relation. Every inverse is found by switching each x and y value. Graphically, inverses are reflections over the line y = x.
f ⁻¹(x) — Inverse Function Notation
If (a, b) is on f, then (b, a) is on f ⁻¹. The domain of f = the range of f ⁻¹, and vice versa.
📊 Numerical — Inverses from Tables
Example 1 — Find the inverse relation from a table
Original
| x | 1 | 3 | 4 | 6 |
|---|---|---|---|---|
| y | −1 | 2 | 0 | 2 |
⇒
Inverse (swap x & y)
| x | −1 | 2 | 0 | 2 |
|---|---|---|---|---|
| y | 1 | 3 | 4 | 6 |
Note: The original table IS a function (each x has one y). The inverse is NOT a function — x = 2 maps to both y = 3 and y = 6.
📈 Graphical — Inverses from Graphs
Steps to Sketch an Inverse Graph (Linear Pieces)
1
List the key points of the original graph in a table
2
Create a new table by switching x and y values
3
Plot the new points and sketch the inverse graph
Example 2 — Key Points: (−2,−1), (−1,2), (2,1), (4,3)
Original key points
| x | −2 | −1 | 2 | 4 |
|---|---|---|---|---|
| y | −1 | 2 | 1 | 3 |
⇒
Inverse points
| x | −1 | 2 | 1 | 3 |
|---|---|---|---|---|
| y | −2 | −1 | 2 | 4 |
Steps to Sketch an Inverse Graph (Nonlinear Pieces)
1
Sketch the line y = x on the graph
2
Points already on y = x stay fixed
3
For other points: switch x and y (or reflect perpendicularly over y = x)
4
Connect the new points following the original pattern
🧮 Analytical — Finding Inverses from Equations
Steps to Find f ⁻¹(x) from an Equation
1
Replace f(x) with y
2
Switch x and y
3
Solve for y (get y by itself)
4
Write the answer as f ⁻¹(x) = ...
Example 3 — f(x) = 3x − 7
y = 3x − 7 → switch: x = 3y − 7
x + 7 = 3y
y = (x + 7) / 3
f ⁻¹(x) = (x + 7) / 3
x + 7 = 3y
y = (x + 7) / 3
f ⁻¹(x) = (x + 7) / 3
For rational functions, use these extra steps after switching x and y:
Extra Steps for Rational Functions
A
Multiply both sides by the denominator to clear the fraction
B
Move all y-terms to one side, everything else to the other
C
Factor out y from the left side
D
Divide both sides by what remains
Example 4a: f(x) = (x−2)/(x+3)
x = (y−2)/(y+3)
x(y+3) = y−2
xy+3x = y−2
xy−y = −3x−2
y(x−1) = −(3x+2)
f ⁻¹(x) = −(3x+2)/(x−1)
x(y+3) = y−2
xy+3x = y−2
xy−y = −3x−2
y(x−1) = −(3x+2)
f ⁻¹(x) = −(3x+2)/(x−1)
Example 4b: f(x) = (2x+1)/(x−3)
x = (2y+1)/(y−3)
x(y−3) = 2y+1
xy−3x = 2y+1
xy−2y = 3x+1
y(x−2) = 3x+1
f ⁻¹(x) = (3x+1)/(x−2)
x(y−3) = 2y+1
xy−3x = 2y+1
xy−2y = 3x+1
y(x−2) = 3x+1
f ⁻¹(x) = (3x+1)/(x−2)
✅ Verifying Inverse Functions
f and g are inverses ⟺ f(g(x)) = x AND g(f(x)) = x
You must show BOTH compositions equal x to prove two functions are inverses.
Example 6 — f(x) = 2x−3 and g(x) = ½x + 32
f(g(x)) = f(½x + 32) = 2(½x + 32) − 3 = (x+3) − 3 = x ✓
g(f(x)) = g(2x−3) = ½(2x−3) + 32 = (x − 32) + 32 = x ✓
g(f(x)) = g(2x−3) = ½(2x−3) + 32 = (x − 32) + 32 = x ✓
Both compositions equal x → they ARE inverses.
Domain Restriction: Sometimes a restricted domain is needed. Example: h(x) = x² + 10 and p(x) = √(x−10) are inverses only when x ≥ 10, which ensures √ gives a non-negative result.
📋 Key Properties of Inverse Functions
- 1g(x) = f ⁻¹(x) — the inverse of f
- 2If (x, y) is on f, then (y, x) is on f ⁻¹
- 3Domain of f = Range of f ⁻¹ | Range of f = Domain of f ⁻¹
- 4A continuous function has an inverse function only if it is strictly increasing or strictly decreasing — it must pass the Horizontal Line Test
✏️ Example 1 (Part II) — Inverses from a Table
f(x) table; g(x) = f ⁻¹(x)
| x | −3 | −2 | 0 | 1 | 4 | 6 |
|---|---|---|---|---|---|---|
| f(x) | 6 | 3 | 1 | −1 | −3 | −7 |
| g(x) | −2 | — | 1 | 4 | 6 | −3 |
a) f(f(0)) = f(1) = −1
b) g(−3) = 4 (g is the inverse: find x where f(x)=−3, that's x=4)
c) g(6) = −3
d) g(g(−1)) = g(1) = 0
e) (f ⁻¹∘f)(−2) = f ⁻¹(3) = g(3) = −2
f) f ⁻¹(−3) = g(−3) = 4
b) g(−3) = 4 (g is the inverse: find x where f(x)=−3, that's x=4)
c) g(6) = −3
d) g(g(−1)) = g(1) = 0
e) (f ⁻¹∘f)(−2) = f ⁻¹(3) = g(3) = −2
f) f ⁻¹(−3) = g(−3) = 4
✏️ Examples 2–4 (Part II) — Inverses from Graphs
Reading f ⁻¹ from a Graph
Key rule: f ⁻¹(a) = b means "find the x-value where f(x) = a." You're doing a reverse lookup on the graph.
Example 2: k defined on [−4, 11]
• Min of k(x) = −3 | Min of k⁻¹(x) = −4 (the least x-value of k)
• k⁻¹(6) = 11 (x-value where k = 6)
• k⁻¹(4) = 6 (x-value where k = 4)
Example 3: f defined on [−2, 8]
• Max of f ⁻¹(x) = 8 (max x-value of f)
• f ⁻¹(3) = 2 | f ⁻¹(1) = 3
• Domain of f ⁻¹ = [−3, 7] = Range of f
• Min of k(x) = −3 | Min of k⁻¹(x) = −4 (the least x-value of k)
• k⁻¹(6) = 11 (x-value where k = 6)
• k⁻¹(4) = 6 (x-value where k = 4)
Example 3: f defined on [−2, 8]
• Max of f ⁻¹(x) = 8 (max x-value of f)
• f ⁻¹(3) = 2 | f ⁻¹(1) = 3
• Domain of f ⁻¹ = [−3, 7] = Range of f
✏️ Example 4 (Part II) — Mixed: Graph + Table
g from graph, h from table
h(x) table
| x | −5 | −1 | 0 | 2 | 5 | 6 |
|---|---|---|---|---|---|---|
| h(x) | −3 | 0 | 3 | 5 | 8 | 10 |
a) g(h(6)) = g(10) = −1 (read from graph of g)
b) g⁻¹(h(0)) = g⁻¹(3) = 4 (x-value on g where g=3)
c) h⁻¹(g(8)) = h⁻¹(0) = −1 (x in h-table where h=0)
d) h⁻¹(g⁻¹(−1)) = h⁻¹(10) = 6 (g⁻¹(−1)=10, then x in table where h=10)
b) g⁻¹(h(0)) = g⁻¹(3) = 4 (x-value on g where g=3)
c) h⁻¹(g(8)) = h⁻¹(0) = −1 (x in h-table where h=0)
d) h⁻¹(g⁻¹(−1)) = h⁻¹(10) = 6 (g⁻¹(−1)=10, then x in table where h=10)
Concept
How do you find an inverse relation from a table?
Tap to reveal ✨
Answer
Switch every x and y value in the table
If (a, b) is in the original, then (b, a) is in the inverse. The inverse may or may not be a function — check if any x-value repeats.
Topic 2.8
Inverse Functions
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Question 1 of 5
The table shows f(x). Which table represents f ⁻¹(x)?
x: 1, 3, 4, 6 | f(x): −1, 2, 0, 2
Question 2 of 5
Find f ⁻¹(x) for f(x) = 3x − 7.
Step 1: y = 3x−7. Step 2: switch to x = 3y−7. Step 3: solve for y.
Question 3 of 5
The graphical property of inverse functions states they are reflections over which line?
Question 4 of 5
f(x) table: f(−3)=6, f(−2)=3, f(0)=1, f(1)=−1, f(4)=−3, f(6)=−7. Let g = f ⁻¹. What is g(−3)?
To find g(−3) = f⁻¹(−3): find the x-value where f(x) = −3
Question 5 of 5
To verify that f(x) = 2x−3 and g(x) = ½x + 32 are inverses, what must you show?
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