2.4 Inverse functions 37
both numerator and denominator have the common factor x, and can be divided by
this factor (whenx 1 ≠ 10 ) without changing the value of the fraction:
EXAMPLE 2.7Simplification of fractions
(1)
(2)
(3)
0 Exercises 17–22
2.4 Inverse functions
Given some functionfand the equationy 1 = 1 f(x), it is usually possible to define, at least
for some values of xand y, a function gsuch thatx 1 = 1 g(y). This new function is the
inverse functionoffand is denoted by the symbolf
− 1(not to be confused with the
reciprocal 12 f):
ify 1 = 1 f(x)thenx 1 = 1 f
− 1(y) (2.7)
EXAMPLE 2.8Ify 1 = 1 f(x) 1 = 12 x 1 + 1 3, findx 1 = 1 f
− 1(y).
To find xin terms of y,
(i) subtract 3 from both sides of the equation: y 1 = 12 x 1 + 131 → 1 y 1 − 131 = 12 x
(ii) divide both sides by 2:
Thereforex 1 = 1 (y 1 − 1 3) 221 = 1 f
− 1(y).
In this example, yis a single-valuedfunction of x; that is, for each value of xthere
exists just one value of y. Similarly, xis a single-valued function of y.
0 Exercises 23–25
EXAMPLE 2.9If , express xin terms of y.
y
ax b
cx d
=
→
−
=
y
x
3
2
ab
aabb
abab
abab
ab
a
22222
−
++
=
+−
++
=
−
()()
()()
()
( ++b)
36
918
31 2
312
12
31 2
2=
=
y
x
y
x
y
x
()
()
()
4
2
22
2
x 2
y
x
y
x
y
=
×
×
=
()
()
xy x
xxy
xy x
xy
yx
y
=
=
2
46
2
223
2
22 3
2()
()()