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4.9. Differentiation of the Inverse Function
Let the domain of the function y = f(x) be an open interval (a, b).
Assume, for certainty, that it is an increasing function (for a decreasing
function the presentation will be quite similar). Such a function
performs a one-to-one correspondence between the domain and the
range. It is a criterion for the inverse function to exist.
Definition 18
Let f be a one-to-one function with domain A and range B.
Then its inverse function h has domain B and range A and is defined
by
h(y)=x ⇔ f(x)=y
for any y in B.
Remark 5
Frequently, the symbol f
−1
is used to denote the inverse function.
The latter “undoes”, so to say, the effect of the original function.
Notice that domain of f
−1
= range of f, range of f
−1
= domain of
f.
Also, the cancellation equations hold:
f
−1
(f(x)) = x for every x in A,
f(f
−1
(x)) = x for every x in B.
Remark 6
How to Find the Inverse Function to a One-To-One Function f.
Step 1. Write y = f(x).
Step 2. Solve this equation for x in terms of y (if possible).
Step 3. To express f
−1
as a function of x, interchange x and y.
The resulting equation is y = f
−1
(x).
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