# Inverse Relation - Definition, Formula & Examples

TL;DR

An inverse relation is what you get when you swap the two coordinates of every ordered pair in a relation — turning each (x,y) into (y,x). This article defines the inverse relation, shows how its graph reflects across the line y=x, proves the inverse relation theorem (R−1)−1=R, and works through six examples — including why an inverse relation is not always an inverse function.

## What Is An Inverse Relation?

An **inverse relation** is the relation obtained by reversing the order of every ordered pair in a given relation. If R is a relation, its inverse is written as R−1, and a pair (a,b) belongs to R exactly when (b,a) belongs to R−1.

R−1={ (b,a): (a,b) ∈ R }.

If R links set A to set B (so R⊆A×B), then R−1 links B back to A (so R−1⊆B×A). The swap is total: every pair flips, no pair is added or dropped.

| Symbol       | Meaning                                                        |
|--------------|----------------------------------------------------------------|
| R            | the original relation (a set of ordered pairs)                 |
| R−1          | the inverse relation                                            |
| (a,b)       | an ordered pair in R                                           |
| A×B         | the set of all pairs with the first element from A and the second from B |

## How Do You Find the Inverse of a Relation?

For a relation listed as ordered pairs, swap each pair's coordinates. For a relation given by an equation, swap x and y and (where useful) solve for y.

### Example 1

**Find the inverse of R={(1,4),(2,5),(3,6)}.**

Swap each pair.

R−1={(4,1),(5,2),(6,3)}.

**Final answer:** R−1={(4,1),(5,2),(6,3)}.

### Example 2

**State the domain and range of R={(a,2),(b,4),(c,1)} and of R−1.**

Original domain ={a,b,c}; original range ={1,2,4}.

Inverse: R−1={(2,a),(4,b),(1,c)}.

Inverse domain ={1,2,4}; inverse range ={a,b,c}.

**Final answer:** the domain and range trade places — the inverse's domain is the original's range, and vice versa.

### Example 3

**A student inverts R={(1,5),(2,5),(3,7)} and concludes the inverse is a function. Is it?**

Wrong attempt. The student swaps to get R−1={(5,1),(5,2),(7,3)} and reasons, "since the original was a function, its inverse must be too."

_Why it breaks:_ Look at the inverse's first coordinates: 5 appears twice, paired with both 1 and 2. A function cannot send one input to two outputs. So the inverse fails the function test even though it is a perfectly valid inverse relation.

_Correct:_ R−1={(5,1),(5,2),(7,3)} is an inverse relation but **not** an inverse function, because the original relation was not one-to-one.

### Example 4

**Find the inverse of the algebraic relation y=x^2.**

Swap the variables.

x=y^2.

Solve for y: y=±√x.

**Final answer:** y=±√x, a valid inverse relation, but the ± shows it is not a function.

### Example 5

**Find the inverse of y=2x+1.**

Swap the variables.

x=2y+1.

Multiply both sides by 2: 2x=2y+1.

Subtract 1: 2x−1=2y.

Divide by 2: y=(2x−1)/2.

**Final answer:** y=(2x−1)/2.

### Example 6

**A relation passes through (0,3), (2,7), and (−1,1). Where does its inverse pass through?**

Swap each pair: the inverse passes through (3,0), (7,2), and (1,−1).

**Final answer:** the inverse passes through (3,0), (7,2), (1,−1), and its graph is the original reflected over the line y=x.

## Where Inverse Relations Show Up

- **Encoding and decoding:** A cipher maps letters to symbols; reading a message back uses the inverse relation.
- **Unit conversion:** Celsius-to-Fahrenheit and its reverse are inverse relations.
- **Databases and lookups:** Any "who has this?" table can be flipped to "what does this person have?" — the same pairs, read backward.

The destination is the inverse function: once a relation is one-to-one, its inverse relation graduates into an inverse function.

## Common Errors With Inverse Relations

### Mistake 1: Assuming the inverse of a function is always a function

**Where it slips in:** Any relation that is not one-to-one.

**The correct way:** Check whether the original is one-to-one. Every relation has an inverse relation; the inverse is a function only when the original passes the horizontal line test.

### Mistake 2: Forgetting to swap domain and range

**Where it slips in:** Stating the domain and range of an inverse.

**The correct way:** The inverse's domain is the original's range, and the inverse's range is the original's domain.

### Mistake 3: Solving an equation before swapping the variables

**Where it slips in:** Finding the inverse of linear relations.

**The correct way:** Swap x and y first, then solve for y.

## Practice Questions

1. Find the inverse of R={(2,8),(3,8),(5,1)} and decide whether the inverse is a function.
2. Find the inverse of y=4x−7.
3. Reflect the points (1,2) and (4,0) across y=x.
4. State the domain and range of R={(p,3),(q,5)} and of R−1.
5. Apply the inverse relation theorem for R={(1,9)}.

## Answers

**Answer to Question 1:** R−1={(8,2),(8,3),(1,5)}. It is **not** a function.

**Answer to Question 2:** y=(x+7)/4.

**Answer to Question 3:** (1,2)→(2,1) and (4,0)→(0,4).

**Answer to Question 4:** Domain={p,q}; range={3,5}; inverse domain={3,5}; inverse range={p,q}.

**Answer to Question 5:** (R−1)−1=R={(1,9)}.

## Conclusion

- An **inverse relation** R−1 is formed by swapping every ordered pair (a,b) in R into (b,a).
- The graph of an inverse relation is the original reflected across y=x.
- The inverse relation theorem says (R−1)−1=R, so inverting twice returns the original.
- An inverse relation is always a relation, but it is a function only when the original is one-to-one.