1.6 Inverse Functions and Logarithms
Inverse Functions
We already know that when we have a function
y = f (x)
We put a value of x into the function and obtain a value of y out of it.
For example:
I go to a store to buy apples. I want to buy 5 bl. apples. How much will
it cost me?
The opposite can also happen.
If we are given the value of y, we would like to know which value of x will
give us this value of y.
Opposite of the above example:
I go to the shop to buy apples. I want to buy $5:00 worth of apples.
What is the weight of apples that I can get from that $5:00?
So we need what is known as an inverse function of f (x), which we can
write as
x = f 1 (y)
Important Note:
f
1
(y) 6=
1
!!!
f (y)
Before we can begin discussing inverse functions, we …rst need to look at
one-by-one functions.
One-to-one function
De…nition 1 A function f is called a one-to-one function if it never takes
on the same function value twice, or more formally
f (x1 ) 6= f (x2 ); whenever x1 6= x2 :
Recall that for a relationship to be de…ned as a function, for every x,
there can only be one corresponding y. All functions, including one-to-one
functions, have this characteristic. They must pass the vertical line test.
One-to-one functions have an extra characteristic, i.e. for every y, there
can only be one corresponding x. One-to-one functions must also pass the
horizontal line test.
1
Theorem 1 (Horizontal Line Test) A function is one-to-one if and only
if no horizontal line intersects its graph more than once.
For example: y = x2 is not a one-to-one function because there are horizontal lines that intersect the graph more than once.
Another example: y = x3 is a one-to-one function because there are no
horizontal lines that intersect the graph more than once.
Some functions are not one-to-one on its entire domain. But it may be a
one-to-one
function on part of its domain.
For example: y = x2 is one-to-one on x 0 or x 2 [0; 1)
2
and y = x2 is also one-to-one on x
0 or x 2 ( 1; 0].
Now suppose that we do not have the graph of a function, and we only
have the algebraic declaration of the function. How do we determine if the
function is one-to-one? We need a proof.
Example 1 Prove that the function f (x) = 3x + 2 is a one-to-one function
in R.
Example 2 Prove that the function g(x) =
[0; 1).
p
x is a one-to-one function in
One-to-one functions are important because they are the class of functions
which have an inverse.
Domain and Range of an inverse function
De…nition 2 Let f be a one-to-one function with domain A and range B.
Then the inverse of f , denoted f 1 , has domain B and range A and is
de…ned by
f
1
(y) = x if and only if f (x) = y, for any y 2 B.
3
Important Note:
The domain of the inverse function f
of f 1 is the domain of f !!!
1
is the range of f , and the range
Example 3 If f (1) = 5, f (3) = 7, and f (8) =
and f 1 (7).
10, …nd f
1
( 10), f
1
(5),
Cancellatin Equations
To determin whether some function g is the inverse of some function f , which
has domain A and range B, one must verify that
g(f (x)) = x for every x 2 A and
f (g(x)) = x for every x 2 B:
If both of the above cancellatin equations are satis…ed, then g = f
1
.
Example 4 Determine whether f (x) = x3 and g(x) = x1=3 are inverse functions.
Suppose, instead, that we are not given a possible canadidate for the
inverse function. How to …nd the inverse function of a function f , if it
exists?
1. Check if f is a one-to-one function over the domain of f . If it is not,
then f has no inverse function. If yes, then goto step 2.
2. Set y = f (x)
3. Rewrite the equation as x in terms of y (if possible)
4. Interchange x and y. This gives us y = f
1
(x).
5. Check to see if the two cancellation equations hold.
4
Example 5 Find the inverse of f (x) = x3 + 2.
Example 6 Find a formula for the inverse of the function.f (x) =
4x 1
:
2x + 3
Graph of the inverse function
The graph of f
line y = x.
1
(x) is obtained by re‡ecting the graph of f (x) about the
Example 7 Sketch the graph of f (x) =
p
x and its inverse.
Example 8 Sketch the graph of f (x) = x3 , x
5
0, and its inverse.
Logarithmic functions
Another way of writing ab = c is loga c = b.
By de…nition, given an exponential function
f (x) = ax ; domain x 2 ( 1; 1) and range y 2 (0; 1)
when a > 0 and a 6= 1, its inverse function is the logarithmic function
f
1
(x) = loga x, domain x 2 (0; 1) and range y 2 ( 1; 1)
The graphs of some logarithmic functions can be found in Chapter 1.6 or
the Reference pages of Stewart’s book.
Example 9 Sketch the graph of f (x) = loga (x) when a > 1. (The most
important logarithmic functions have base a > 1:)
Example 10 Evaluate log10 (0:001).
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Natural Logarithms
When the base of a logarithm is e
2:71828 , it is called the natural
logarithm.
Notation: loge x = ln x
ln x and ex are the inverse function of each other:
ln x = y , ey = x
Laws of logarithms
Theorem 2 (Laws of logarithms) If x and y are positive numbers, a > 0
and a 6= 1, then
1. loga (xy) = loga x + loga y
2. loga
x
= loga x
y
loga y
3. loga xr = r loga x for any real number r
4. For any real number b such that b > 0 and b 6= 1, loga x =
particular, loga x =
logb x
. In
logb a
ln x
. This is the Changes of Base Formula.
ln a
Some other properties of logarithms:
1. loga ax = x, for any x 2 ( 1; 1)
2. in particular, loga a = 1
3. aloga x = x, for any x 2 (0; 1)
4. loga 1 = 0 since a0 = 1
The properties above are also true for natural logrithms:
ln ex = x for any x 2 ( 1; 1)
ln e = 1
eln x = x for any x 2 (0; 1)
ln 1 = 0
Example 11 Use the Laws of Logarithms to evaluate log2 80
7
log2 5.
Example 12 Find the exact value of each expression
(1) ln 1e
(2) log10
(3) e
p
10
2 ln 5
10
(4) ln ln ee
Example 13 Rewrite log8 5 in terms of natural logrithms.
Example 14 Express ln a + 12 ln b as a single logrithm:
Example 15 Find x if ln x = 5.
Example 16 Solve the equation e5
3x
8
= 10 for x.
Example 17 Determine the inverse function of y =
Example 18 Sketch the graph of y = ln(x
formations.
2)
ex
1 + 2ex
1 using elementary trans-
Inverse Trigonometric Functions
The output of inverse trig. functions is always an angle.
Suppose we are given the trig. function f (x) = sin x as shown in the
following graph:
The trig. function is not a one-to-one function. For it to be a one-to-one
function, we need to restrict its domain, usually to
x
, since
2
2
f (x) = sin x is monotone on this interval and it has both positive and negative
values on this interval.
Then we can de…ne its inverse function as
9
De…nition 3 (Inverse Sine Function) The inverse sine function, denoted
by sin 1 , is de…ned to be y = sin 1 (x) if and only if x = sin(y) and
2
.
y
2
The inverse sine function is also called arcsine function, which is denoted
by arcsin.
Important note:
1
sin 1 (x) 6=
sin(x)
Example 19 State the domain of f (x) = arcsin x.
Example 20 State the range of f (x) = arcsin(x).
Example 21 Sketch the graph of f (x) = sin 1 (x) = arcsin(x).
De…nition 4 (Inverse Cosine Function) The inverse cosine function, denoted by cos 1 , is de…ned to be y = cos 1 (x) if and only if x = cos(y) and
0 y
.
The inverse cosine function is also called arccosine function, which is
denoted by arccos.
Example 22 State the domain of f (x) = arccos x.
10
Example 23 State the range of f (x) = arccos(x).
Example 24 Sketch the graph of f (x) = cos 1 (x) = arccos(x).
De…nition 5 (Inverse Tangent Function) The inverse tangent function,
denoted by tan 1 , is de…ned to be y = tan 1 (x) if and only if x = tan(y) and
<x< .
2
2
The inverse tangent function is also called arctangent function, which is
denoted by arctan.
Example 25 State the domain of f (x) = arctan x.
Example 26 State the range of f (x) = arctan(x).
Example 27 Sketch the graph of f (x) = tan 1 (x) = arctan(x).
11
De…nition 6 (Inverse Cotangent Function) The inverse tangent function, denoted by cot 1 , is de…ned to be y = cot 1 (x) if and only if x = cot(y)
and 0 < x < .
The inverse cotangent function is also called arccotangent function, which
is denoted by arccot.
Example 28 State the domain of f (x) = arccot x.
Example 29 State the range of f (x) = arccot(x).
Example 30 Sketch the graph of f (x) = cot 1 (x) = arccot(x).
De…nition 7 (Inverse Secant Function) The inverse secant function, denoted by sec 1 , is de…ned to be y = sec 1 (x) if and only if x = sec(y) and
y 2 [0; 2 ) [ [ ; 32 ).
The inverse secant function is also called arcsecant function, which is
denoted by arcsec.
Example 31 State the domain of f (x) = arcsec x.
12
Example 32 State the range of f (x) = arcsec(x).
De…nition 8 (Inverse Cosecant Function) The inverse cosecant function,
denoted by csc 1 , is de…ned to be y = csc 1 (x) if and only if x = csc(y) and
y 2 (0; 2 ] [ ( ; 32 ].
The inverse cosecant function is also called arccosecant function, which
is denoted by arccsc.
Example 33 State the domain of f (x) = arccsc x.
Example 34 State the range of f (x) = arccsc(x).
Example 35 Evaluate sin
1
1
2
:
Example 36 Evaluate tan(arcsin 13 ):
Example 37 Simplify the expression cos (tan
13
1
x) :