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**Unformatted text preview: **y are switched. Similarly, the horizontal asymptote y = −2 of the
graph of g corresponds to the vertical asymptote x = −2 of the graph of g −1 . 5.2 Inverse Functions 303
y
6
5
4 y=x 3
2
1 −6 −5 −4 −3 −2 −1 1 2 3 4 5 x 6 −1
−2
−3
−4
−5
−6 y = g (x) and y = g −1 (x) We now return to f (x) = x2 . We know that f is not one-to-one, and thus, is not invertible.
However, if we restrict the domain of f , we can produce a new function g which is one-to-one. If
we deﬁne g (x) = x2 , x ≥ 0, then we have
y y 7
6 6 5 5 4 4 3 3 2 2 1 −2 −1 7 1 1 2 −2 −1 x
restrict domain to x ≥ 0 y = f (x) = x2 −− − − − − − −
− − − − − − −→ 1 2 x y = g (x) = x2 , x ≥ 0 The graph of g passes the Horizontal Line Test. To ﬁnd an inverse of g , we proceed as usual
y
y
x
y
y =
=
=
=
= g (x)
x2 , x ≥ 0
y 2 , y ≥ 0 switch x and y
√
±x
√
x
since y ≥ 0 304 Further Topics in Functions √
We get g −1 (x) = x. At ﬁrst it looks like we...

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