If y = f (x),
then we can write x = f −1(y) where f
−1 is the inverse of f. If y = f (x) is a
one-to-one function, then f −1(y) is also a one-to-one
function. In this case, x = f −1 (f (x)) = f
(f −1(x)) for values of x where both f (x)
and f −1(x) are defined. For example ln x, which
maps R+ to R is the inverse of ex. x = eln
x = ln (ex) for all x ∈ R+.
(Note the x ∈ R+ ensures that ln x is defined.).
If y = f (x)
is a many-to-one function, then x = f −1(y) is a
one-to-many function. f −1(y) is a multi-valued
function. We have x = f (f −1(x)) for values of
x where f −1(x) is defined, however x ≠ f −1(f (x)).
There are diagrams showing one-to-one, many-to-one and one-to-many functions in
Figure 1.2.
Example y = x2, a
many-to-one function has the inverse x = y1/2. For
each positive y, there are two values of x such that x = y1/2.
y = x2 and y = x1/2 are graphed in Figure
1.3.
We say that there are two branches
of y = x1/2 : the positive and the negative branch. We denote
the positive branch as y = √x; the negative branch is y =
−√x. We call √x the principal branch of x1/2.
Note that √x is a one-to-one function. Finally, x = (x1/2)2
since (±√x)2 = x, but x ≠ (x2)1/2
since (x2)1/2 = ±x. y = √x is
graphed in Figure 1.4.
domain range domain range
one-to-many
domain range
Figure 1.2:
Diagrams of One-To-One, Many-To-One and One-To-Many Functions
Figure 1.3: y = x2
and y = x1/2
Figure 1.4: y
= √x
Now consider the many-to-one
function y = sin x. The inverse is x = arcsin y.
For each y ∈ [−1..1] there are an infinite number of values x such
that x = arcsin y. In Figure 1.5 is a graph of y = sin x
and a graph of a few branches of y = arcsin x.
Figure 1.5: y = sin x
and y = arcsin x
Example arcsin x
has an infinite number of branches. We will denote the principal
branch by Arcsin x which maps [−1..1] to
. Note that x = sin (arcsin x), but
x ≠
arcsin (sin x). y = Arcsin x in Figure 1.6.
. Note that x = sin (arcsin x), but
x ≠
arcsin (sin x). y = Arcsin x in Figure 1.6.
Figure 1.6: y
= Arcsin x
Example Consider
11/3. Since x3 is a one-to-one function, x1/3
is a single-valued function. (See Figure 1.7.) 11/3 =
1.
Example Consider
arccos(1/2). cos x and a portion of arccosx are graphed in Figure
1.8. The equation cos x = 1/2 has the two solutions x = ±π/3
in the range x ∈ (−π..π]. We use the periodicity of the cosine,
Figure 1.7: y = x3
and y = x1/3
cos(x + 2π) = cos x, to
find the remaining solutions.
arccos(1/2) = {±π/3 +
2nπ}, n ∈Z.
Figure 1.8: y = cos x
and y = arcos x
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