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Functions and Inverses

This guide explores the concept of inverse relations in mathematics. We define an inverse relation as one that "undoes" the original relationship, denoted by f⁻¹(x). For instance, if f(x) = x - 5, then its inverse is f⁻¹(x) = x + 5. We examine the domains and ranges of functions and their inverses, explaining how the domain of a function's inverse corresponds to the range of the original function. Additionally, we provide examples and exercises on finding the inverses of various relations and functions to enhance understanding.

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Functions and Inverses

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  1. Functions and Inverses

  2. Inverses An INVERSE RELATION “undoes” the relation. The Inverse of f(x) is denoted by f-1(x)

  3. Inverse So if f(x) = x – 5 Then f-1(x) = x + 5

  4. Domains and Ranges The domain of a relation’s inverse is its range, and the range of a relation’s inverse is its domain. f(x): D R f-1(x): D R -1 2 6 7 -1 2 6 7 1 2 4 1 2 4

  5. More Inverse Find the inverse of the following relation: {(2,3), (4,5), (1,3)}

  6. Try some! Find the inverse of the following relations: • {(3,4), (-4, -6), (-3, 2), (6, -1)} • {(2, 5), (1, 5), (3, 5)}

  7. Functionality of Inverses If a relation is a function, does it’s inverse have to be??

  8. How to find the inverse of a function: Switch the x and y, and solve for y. Ex: y = 5x - 15

  9. Find the inverse: Ex: y = 6 – 2x

  10. You Try! Find the Inverse of the following: • y = x – 5 • y = 2x + 10 • y = 4x + 5

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