Unformatted text preview: r a proof of this, we refer the reader to Example 1.1.6 in Section 1.1 and Exercise
14 in Section 2.1. A plot of the inverse functions f (x) = 3x + 4 and g (x) = x−4 conﬁrms this.
2 y = f (x) y=x 1 x
−2 −1 1 2 −1 y = g (x)
−2 If we abstract one step further, we can express the sentiment in Deﬁnition 5.2 by saying that f and
g are inverses if and only if g ◦ f = I1 and f ◦ g = I2 where I1 is the identity function restricted1
to the domain of f and I2 is the identity function restricted to the domain of g . In other words,
I1 (x) = x for all x in the domain of f and I2 (x) = x for all x in the domain of g . Using this
description of inverses along with the properties of function composition listed in Theorem 5.1,
The identity function I , which was introduced in Section 2.1 and mentioned in Theorem 5.1, has a domain of all
real numbers. In general, the domains of f and g are not all real numbers, which necessitates the restrictions listed
here. 5.2 Inverse Functions 295 we can show that f...
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