Linear Algebra For Dummies
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If you’re given a function and must find its inverse, first remind yourself that domain and range swap places in the functions. Literally, you exchange f(x) and x in the original equation. When you make that change, you call the new f(x) by its true name — f–1(x) — and solve for this function.

For example, follow the steps to find the inverse of this function:

The function y equals two x minus one, divided by three.
  1. Switch f(x) and x.

    When you switch f(x) and x, you get

    Switching x and y in a function.

    (Note: To make the notation less clumsy, you can rewrite f(x) as y and then switch x and y.)

  2. Change the new f(x) to its proper name — f–1(x).

    The equation then becomes

    Changing y's name to its proper name.
  3. Solve for the inverse.

    This step has three parts:

    1. Multiply both sides by 3 to get 3x = 2f–1(x) –1.

    2. Add 1 to both sides to get 3x + 1 = 2f–1(x).

    3. Lastly, divide both sides by 2 to get your inverse:

      The new inverse function.

About This Article

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Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies.

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