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Understanding Inverse Functions and Graph Relationships

Inverse functions are formed by interchanging the input and output of a relation. To graph the inverse, swap the coordinates of the original relation. This results in a reflection of the graph about the line y=x. To determine if a function has an inverse, use the vertical and horizontal line tests. Verification of inverses requires composing the functions and checking if they result in the identity function. This guide will explore how to find inverses, graph them, and discuss the nature of functions and their inverses.

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Understanding Inverse Functions and Graph Relationships

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  1. Inverse Graphs

  2. Inverse • The inverse of a relation is formed by interchanging the variables.

  3. Interchange to get inverse graph • Interchange x and y to get the inverse function and graph it along with the original relation.

  4. Answer • Interchange columns and plot on original graph

  5. Tests • Inverses are reflections about the line y=x • The vertical line test is for determining whether a relation is a function. • The horizontal line test is for determining whether a relation has an inverse function.

  6. Functions and inverse functions • Are the following functions? Do they have inverse functions?

  7. Answer • Yes (passes vertical), no (fails horizontal) • Yes (passes vertical),yes (passes horizontal) • No (fails vertical), yes (passes horizontal)

  8. Verifying inverses • To verify that two relations are inverses of one another, compose each and get the identity x both times.

  9. Verifying • Verify that

  10. Answer • Compose both ways.

  11. To find an inverse • Switch the variables and solve for y

  12. Finding Inverses • Find the inverse of f(x)=3x-5.

  13. Answer • Switch x and y and solve for y.

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