Inverse of invertible functions
The inverse function of is denoted by .
Not to be confused with reciprocal functions in HL.
When we are solving for in an equation, we are really finding the inverse function.
Contents
Properties
The defining property of an inverse function:
An invertible function satisfies
The domain of is the range of ; the range of is the domain of . You can use this without finding the inverse.
The inverse function is the mirror image of the invertible function across the diagonal line .
An invertible function and its inverse can only meet on the diagonal line.
The inverse is also invertible, such that
Calculating the inverse function
Remember that inverse function “undoes” the effect of the original function. That is, solve for in . You can either solve for then switch and , or you can switch first then solve for ; it is personal preference which order you do it in.
Example: Find given that
It can be shown that
which is much easier to work with. Let’s switch variables and solve for .
Note that in the original function, the point is not part of the graph, so is not part of the graph of .
Our final answer is
In SL, when ask to find the inverse function, the given function will be invertible. This is not necessarily the case in HL. See next page for additional steps HL candidates should do.
Calculator
Graph inverse relations on TI-84 Plus and this is not inverse functions on TI-84 Plus