Pre-Calculus 12 Notes

By: Gen L.

Lesson 5: Inverse of A Relation

The Inverse

To Inverse is to undo or reverse a position, order or effect.

• A Transformation
• Graph is the reflection along the line
• Domain: Range of , Range: Domain of
• If is also a function, we can use the notation

Graphing an inverse

Invariant Points

• Like other transformations, the Inverse may have Invariant points.
• Any point are Invariant.
• In the previous example: is invariant.

Domain and Range

• For the previous example:
  • Domain of :
  • Range of :
• The inverse flips the points.
  • Domain of :
  • Range of :

The Important Thing

• In the previous example, was the inverse a function?
  • No! The inverse fails the VLT.
• In order to be a function: if a vertical line passes through two values at any point on a curve, it is not a function.
  • This is the Vertical Line Test.
• Likewise, if a horizontal line passes through two values at any point on a function, the inverse is not a function.
  • This is the Horizontal Line Test.

Restrictions

For example, take , and graph its inverse.

• Is it a function?
  • No! The original failed the HLT.
• We need to restrict the domain so that the inverse is a function.
• There's really one main choice: or ,
• hmm... this looks familiar.

Determining Equations

Answers:

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