To find the inverse of a relation, you essentially swap the roles of the input and output variables. Here's a breakdown of how to do it:
Understanding Inverse Relations
An inverse relation "undoes" the original relation. If a relation maps x to y, the inverse relation maps y back to x.
Steps to Find the Inverse
The core idea is to interchange the x and y values. Here's how it's done:
- Replace every x with y and every y with x in the equation that describes the relation. This is according to the provided reference.
- Solve for y. This will give you the equation of the inverse relation in the standard "y = " form.
Examples
Let's illustrate this with some examples:
Example 1: Simple Linear Equation
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Original Relation: y = 2x + 3
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Step 1: Swap x and y: x = 2y + 3
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Step 2: Solve for y:
- x - 3 = 2y
- y = (x - 3) / 2
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Inverse Relation: y = (x - 3) / 2
Example 2: Another Linear Equation
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Original Relation: x - 5y = 10
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Step 1: Swap x and y: y - 5x = 10
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Step 2: Solve for y:
- y = 5x + 10
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Inverse Relation: y = 5x + 10
Example 3: Relation defined as a Set of Ordered Pairs
Consider the relation: {(1, 2), (3, 4), (5, 6)}
To find the inverse, simply swap the x and y values in each ordered pair:
- Inverse Relation: {(2, 1), (4, 3), (6, 5)}
Visualizing Inverse Relations
Graphically, the inverse of a relation is a reflection of the original relation across the line y = x. This reflection visually represents the swapping of x and y values.
Important Considerations
- Not all relations have inverses that are functions. For the inverse to be a function, it must pass the vertical line test.
- The domain of the inverse relation is the range of the original relation, and the range of the inverse relation is the domain of the original relation.
