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Unformatted text preview: s is, after all, how they
were deﬁned.) Our last example solidiﬁes this and reviews all of the material in the section.
Example 6.1.5. Let f (x) = 2x−1 − 3.
1. Graph f using transformations and state the domain and range of f .
2. Explain why f is invertible and ﬁnd a formula for f −1 (x).
3. Graph f −1 using transformations and state the domain and range of f −1 .
4. Verify f −1 ◦ f (x) = x for all x in the domain of f and f ◦ f −1 (x) = x for all x in the
domain of f −1 .
5. Graph f and f −1 on the same set of axes and check the symmetry about the line y = x.
1. If we identify g (x) = 2x , we see f (x) = g (x − 1) − 3. We pick the points −1, 2 , (0, 1)
and (1, 2) on the graph of g along with the horizontal asymptote y = 0 to track through
the transformations. By Theorem 1.7 we ﬁrst add 1 to the x-coordinates of the points on
the graph of g (shifting g to the right 1 unit) to get 0, 2 , (1, 1) and (2, 2). The horizontal
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