Unformatted text preview: from g and returns them to their respective
inputs. We now have enough background to state the central deﬁnition of the section.
Definition 5.2. Suppose f and g are two functions such that
1. (g ◦ f )(x) = x for all x in the domain of f and
2. (f ◦ g )(x) = x for all x in the domain of g .
Then f and g are said to be inverses of each other. The functions f and g are said to be
Our ﬁrst result of the section formalizes the concepts that inverse functions exchange inputs and
outputs and is a consequence of Deﬁnition 5.2 and the Fundamental Graphing Principle for Functions.
Theorem 5.2. Properties of Inverse Functions: Suppose f and g are inverse functions.
• The rangea of f is the domain of g and the domain of f is the range of g
• f (a) = b if and only if g (b) = a
• (a, b) is on the graph of f if and only if (b, a) is on the graph of g
a Recall this is the set of all outputs of a function. The third property in Theorem 5.2 tells us that the graphs of inverse functions are reﬂections about
the line y = x. Fo...
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