The concept of an inverse function, its graph as a reflection in the line y=x, finding the inverse of a specific value, and domain and range of inverse functions. When inverse functions exist, and how to find them when they do.
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We can find x=f−1(b) by applying the function to both sides:
So finding f−1(b) is equivalent to solving f(x)=b.
Graphically, find f−1(b) is equivalent to being given y=b, and finding the value of x for which that is true:
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The graph of a function f shows all the points (x,f(x)). Since f−1 undoes f, its graph will show all the points (f(x),x). In other words, the x and y values are swapped.
This is equivalent to reflecting the curve y=f(x) in the line y=x:
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Since f−1 undoes f, the domain of f−1 is all the possible values f could output. That is, the domain of f−1 is the range of f.
The range of f−1 is all the possible values that could have gone into f. Thus, the range of f−1 is the domain of f.
Formally, the inverse function is such that
We call x the identity function, as I(x)=x composed with any function gives the same function.
Inverse functions f−1(x) can be found algebraically by switching x and y in the expression for f(x) and attempting to isolate y.
Recall that a function f must pass the vertical line test to guarantee that each input gives at most one output.
Since f−1 is also a function, it too must pass the vertical line test. But since the graph of f−1(x) is a reflection of the graph of f(x) in the line y=x, the graph of f(x) must then pass the horizontal line test:
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A function that passes the horizontal line test is said to be one to one - each input yields exactly one distinct output. Such a function is said to be invertible, which means f−1 exists.
A function is said to be self-inverse if it is its own inverse:
Since the graph of f−1(x) is the mirror image of f(x) in the line y=x, the graph of a self-inverse function must be symmetric in the line y=x.
When a function f is not one to one, ie it does not pass the horizontal line test, its domain needs to be restricted for the inverse to exist.