Asked by hg
To find the inverse, exchange the coordinates of the ordered pairs.
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Answered by
GPT-5 mini
AI
Yes — for a relation given as ordered pairs, the inverse relation is formed by swapping each pair’s coordinates: (x, y) becomes (y, x).
Quick points and examples:
- Example (pairs): If R = {(1, 2), (3, 4), (5, 0)}, then R⁻¹ = {(2, 1), (4, 3), (0, 5)}.
- Algebraic functions: To find the inverse of y = f(x), swap x and y (write x = f(y)) and then solve that equation for y. Example: y = 2x + 3 → swap → x = 2y + 3 → solve → y = (x − 3)/2, so f⁻¹(x) = (x − 3)/2.
- Graphically: the inverse is the reflection of the graph across the line y = x.
- Domain and range swap: domain(f⁻¹) = range(f) and range(f⁻¹) = domain(f).
- Not every inverse is a function: f has an inverse function only if it is one-to-one (horizontal line test). If the inverse relation contains duplicate first coordinates, it’s not a function unless you restrict the domain.
If you have a specific relation or function you want inverted, give it and I’ll compute the inverse.
Quick points and examples:
- Example (pairs): If R = {(1, 2), (3, 4), (5, 0)}, then R⁻¹ = {(2, 1), (4, 3), (0, 5)}.
- Algebraic functions: To find the inverse of y = f(x), swap x and y (write x = f(y)) and then solve that equation for y. Example: y = 2x + 3 → swap → x = 2y + 3 → solve → y = (x − 3)/2, so f⁻¹(x) = (x − 3)/2.
- Graphically: the inverse is the reflection of the graph across the line y = x.
- Domain and range swap: domain(f⁻¹) = range(f) and range(f⁻¹) = domain(f).
- Not every inverse is a function: f has an inverse function only if it is one-to-one (horizontal line test). If the inverse relation contains duplicate first coordinates, it’s not a function unless you restrict the domain.
If you have a specific relation or function you want inverted, give it and I’ll compute the inverse.
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