Unformatted text preview: notation persist? As we mentioned in Section 5.1, the identity function I is to function composition
what the real number 1 is to real number multiplication. The choice of notation f −1 alludes to the
property that f −1 ◦ f = I1 and f ◦ f −1 = I2 , in much the same way as 3−1 · 3 = 1 and 3 · 3−1 = 1.
Let’s turn our attention to the function f (x) = x2 . Is f invertible? A likely candidate for the inverse
is the function g (x) = x. Checking the composition yields (g ◦ f )(x) = g (f (x)) = x2 = |x|, which
is not equal √ x for all x in the domain (−∞, ∞). For example, when x = −2, f (−2) = (−2)2 = 4,
but g (4) = 4 = 2, which means g failed to return the input −2 from its output 4. What g did,
however, is match the output 4 to a diﬀerent input, namely 2, which satisﬁes f (2) = 4. This issue
is presented schematically in the picture below.
f x = −2
4 x=2 g
We see from the diagram that since both f (−2) and f (2) are 4, it is impossible to construct a
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