You really do need to remember all the "inverse stuff."

Facts about Inverse Functions:

f ^-1(f(x)) = f(f ^-1(x)) = x

The inverse of the function equals the function of the inverse.

The inverse and the function undo each other resulting

    in the original number.

To take an inverse algebraically:

1st: Use the symbol y in place of f(x).

2nd: Exchange the x for y and the y for x and solve for y.

3rd: Use the old domain as the new range,

      since x and y have been exchanged.

4th: Use the old range as the new domain.

      This may be very important since many functions are not 1-1, invertable. If an inverse is REALLY needed, the domain must be restricted or unique inverses are not possible.

5th: Restate the inverse with these 3 parts:

      f^-1(x) =

      the expression from step 2

      , the restriction on the domain.

To take the inverse graphically,

1st: Graph the function.

2nd: To exchange the x for y and the y for x, first graph

      the line y = x which will act as a mirror to

      exchange the variables.

3rd: For each point (x,y) on the original function,

      plot the point (y,x) on the inverse.

4th: You must now write an expression for the graph

      you've drawn.

      Use a basic function such as f(x) = x² or f(x) = x.

      Use curve shifting coding such as f(x) = f(x - h) + k

      to "place" the function h to the right or left and

      to "place" the function k up or down.

5th: Use the old domain as the new range,

      since x and y have been exchanged.

6th: Use the old range as the new domain.

7th: Restate the inverse with these 3 parts:

      f^-1(x) =

      the expression from step 4

      , the restriction on the domain.

 

      To try this technqiue some functions have been graphed. If you click on the graph, the inverse function and its graph are displayed. See sample functions & graphs, then their inverses and graphs.

      Since the trig functions are periodic they are not 1-to-1 and to have an inverse, the domain of each must be restricted.

      The restriction on the domain of the original function was selected so that the arcfunction returned unique values for each domain value and provided for negatives and positives.

Examples & other notes
• 2/4 -function's reciprocal & inverse.
• Take the inverse of 3 different functions
• 2/7 - Take an inverse function:
  using words or order of operations, graphically, or symbolically
  2nd example
• 2/12 - algebraically & graphically taking an inverse