College Algebra, 2013a

386 Further Topics in Functions
y
2
y = x
1
−4 −3 −2 −1 1 2 3 4
−1
y = f(x)
−2
y = f −1 (x)
x
2. To find g −1 (x), we start with y = g(x). We note that the domain of g is (−∞, 1) ∪ (1, ∞).
y = g(x)
2x
y =
1 − x
2y
x =
switch x and y
1 − y
x(1 − y) = 2y
x − xy = 2y
x = xy +2y
x = y(x + 2) factor
y =
x
x +2
We obtain g −1 (x) =
x
x+2 . To check this analytically, we first check ( g −1 ◦ g ) (x) =x for all x
in the domain of g, that is, for all x ≠1.
(
g −1 ◦ g ) (x) = g −1 (g(x))
( ) 2x
= g −1 1 − x
( 2x
)
=
=
1 − x
( ) 2x
+2
1 − x
( ) 2x
1 − x (1 − x)
( ) · 2x (1 − x)
+2
1 − x
clear denominators