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Every one-to-one function f has an inverse function f ^{-1} which essentially reverses the operations performed by f .
More formally, if f is a one-to-one function with domain D and range R , then its inverse f ^{-1} has domain R and range D . f ^{-1} is related to f in the following way: If f (x) = y , then f ^{-1}(y) = x . Written another way, f ^{-1}(f (x)) = x .
Example:
f (x) = 3x - 4
. Find
f
^{-1}(x)
.
The procedure for finding
f
^{-1}(x)
from
f (x)
involves first solving for
x
in terms
of
y
.
y | = 3x - 4 | ||
x | = |
Now switch the variables x and y in the equation to generate the inverse:
y | = | ||
f ^{-1}(x) | = |
A function and its inverse are related geometrically in that they are reflections about the line y = x :
Thus, if (a, b) is a point on the graph of f , then (b, a) is a point on the graph of f ^{-1} .
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