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Unformatted text preview: unction inverses are unique.2 Suppose g and h are both inverses of a function
f . By Theorem 5.2, the domain of g is equal to the domain of h, since both are the range of f .
This means the identity function I2 applies both to the domain of h and the domain of g . Thus
h = h ◦ I2 = h ◦ (f ◦ g ) = (h ◦ f ) ◦ g = I1 ◦ g = g , as required.3 We summarize the discussion of the
last two paragraphs in the following theorem.4
Theorem 5.3. Uniqueness of Inverse Functions and Their Graphs : Suppose f is an
• There is exactly one inverse function for f , denoted f −1 (read f -inverse)
• The graph of y = f −1 (x) is the reﬂection of the graph of y = f (x) across the line y = x.
The notation f −1 is an unfortunate choice since you’ve been programmed since Elementary Algebra
to think of this as f . This is most deﬁnitely not the case since, for instance, f (x) = 3x + 4 has as
its inverse f −1 (x) = x−4 , which is certainly diﬀerent than f (x) = 3x1 . Why does this confusing
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