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Concept

Inverse of a Function

Every function has an inverse relation. If this inverse relation is also a function, then it is called an inverse function. In other words, the inverse of a function is another function such that they undo each other.

Also, if is the input of a function and its corresponding output, then is the input of and its corresponding output.

Example

Consider a function and its inverse
These functions will be shown to undo each other. To do so, it needs to be proven that and that To start, the first equality will be proven. First, the definition of will be used.
Now, in the above equation, will be substituted for
Simplify left-hand side
A similar procedure can be performed to show that
Definition of First Function Substitute Second Function Simplify

Therefore, and undo each other. The graphs of these functions are each other's reflection across the line This means that the points on the graph of are the reversed points on the graph of

Graph of the linear function f(x)=2*x-3 with a point (a,b) on it and the graph of its inverse function f^{-1}(x)=x/2+3/2 with a point (b,a) on it are depicted. The line of symmetry x=y for these functions is also shown.