Inverse Relation - Definition, Formula & Examples
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
- Find the inverse of R={(2,8),(3,8),(5,1)} and decide whether the inverse is a function.
- Find the inverse of y=4x−7.
- Reflect the points (1,2) and (4,0) across y=x.
- State the domain and range of R={(p,3),(q,5)} and of R−1.
- 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.