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In addition to the revision notes for Inverse Functions on this page, you can also access the following Functions learning resources for Inverse Functions
| Tutorial ID | Title | Tutorial | Video Tutorial | Revision Notes | Revision Questions |
|---|---|---|---|---|---|
| 16.6 | Inverse Functions |
In these revision notes for Inverse Functions, we cover the following key points:
By definition, an inverse function - otherwise known as an anti-function - is defined as a function, which can reverse into another function.
Put simply, an inverse function is a function that undoes the action of another function.
Not all functions have an inverse. In general, only injective (one-to-one) functions have an inverse. This is the first condition for a function to have an inverse.
The other condition concern the relationship between the domain and range. The following rules apply to the domain and range in inverse functions:
There are two analytical methods used to find the inverse of a function but that have the same procedure and obviously, provide the same output.
The first method consists of expressing f(x) (or y) in terms of x and swapping the places of x and f(x) at the end of the procedure.
The second method consists of swapping the position of x and f(x) immediately; then we continue with the rest of the operations. For convenience, we can express f(x) by y, as they mean the same thing. Let's explain this method through an example.
The graph of a function f(x) and that of its inverse are symmetrical with respect to the line y = x. This property is used for identifying the inverse of a function f(x) when its graph is given.
Another method to identify the inverse of a function shown graphically is to first the formula of the original function from the figure and then use any of the analytical methods covered in this tutorial for identifying the formula of the inverse function. This avoids the necessity to use the symmetry line y = x in the process.
For certain restricted domains, even the quadratic functions (as well as other types of functions that normally don't have an inverse) become one-to-one.
Therefore, they may have an inverse f - 1(x) given the condition that the domain of f(x) must be the range of f - 1(x) and the range of f(x) must be the domain of f - 1(x).Other functions that have an inverse include the exponential and logarithmic functions, trigonometric functions, reciprocal functions, etc. For example, an exponential function will have a logarithmic one as an inverse and vice-versa.
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