FUNCTIONS AND MODELS
1.6 Inverse Functions and Logarithms
INVERSE FUNCTIONS
The table gives data from an experiment
in which a bacteria culture started with
100 bacteria in a limited nutrient medium.
ƒ The size of the bacteria
population was recorded
at hourly intervals.
ƒ The number of bacteria
N is a function of the
time t: N = f(t).
INVERSE FUNCTIONS
INVERSE FUNCTIONS
However, suppose that the biologist changes
her point of view and becomes interested in
the time required for the population to reach
various levels.
This function is called the inverse
function of f.
ƒ It is denoted by f -1 and read “f inverse.”
ƒ In other words, she is thinking of t
as a function of N.
INVERSE FUNCTIONS
INVERSE FUNCTIONS
Thus, t = f -1(N) is the time required for
the population level to reach N.
The values of f -1can be found by reading
the first table from right to left or by consulting
the second table.
ƒ For instance, f -1(550) = 6, because f(6) = 550.
1
INVERSE FUNCTIONS
INVERSE FUNCTIONS
Not all functions possess
inverses.
Note that f never takes on the same
value twice.
ƒ Let’s compare the functions f and g whose
arrow diagrams are shown.
ƒ Any two inputs in A have different outputs.
INVERSE FUNCTIONS
INVERSE FUNCTIONS
However, g does take on the same
value twice.
In symbols, g(2) = g(3)
ƒ Both 2 and 3 have the same output, 4.
but
f(x1) ≠ f(x2) whenever x1 ≠ x2
INVERSE FUNCTIONS
ONE-TO-ONE FUNCTIONS
Functions that share this property
with f are called one-to-one functions.
A function f is called a one-to-one
function if it never takes on the same
value twice.
That is,
f(x1) ≠ f(x2)
Definition 1
whenever x1 ≠ x2
2
ONE-TO-ONE FUNCTIONS
ONE-TO-ONE FUNCTIONS
If a horizontal line intersects the graph of f
in more than one point, then we see from
the figure that there are numbers x1and x2
such that f(x1) = f(x2).
So, we have the following
geometric method for determining
whether a function is one-to-one.
ƒ This means f is
not one-to-one.
HORIZONTAL LINE TEST
ONE-TO-ONE FUNCTIONS
A function is one-to-one if and only if
no horizontal line intersects its graph
more than once.
Is the function
f(x) = x3
one-to-one?
ONE-TO-ONE FUNCTIONS
ONE-TO-ONE FUNCTIONS
E. g. 1—Solution 1
If x1 ≠ x2, then x13 ≠ x23.
ƒ Two different numbers can’t have the same cube.
Example 1
E. g. 1—Solution 2
From the figure, we see that no horizontal
line intersects the graph of f(x) = x3 more
than once.
ƒ So, by the Horizontal Line Test, f is one-to-one.
ƒ So, by Definition 1, f(x) =
x3
is one-to-one.
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ONE-TO-ONE FUNCTIONS
Example 2
Is the function
g(x) = x2
one-to-one?
ONE-TO-ONE FUNCTIONS
E. g. 2—Solution 1
The function is not one-to-one.
ƒ This is because, for instance,
g(1) = 1 = g(-1)
and so 1 and -1 have the same output.
ONE-TO-ONE FUNCTIONS
E. g. 2—Solution 2
From the figure, we see that there are
horizontal lines that intersect the graph
of g more than once.
ƒ So, by the Horizontal Line Test, g is not one-to-one.
ONE-TO-ONE FUNCTIONS
Definition 2
Let f be a one-to-one function with
domain A and range B.
Then, its inverse function f -1 has domain B
and range A and is defined by
f −1 ( y ) = x
⇔
ONE-TO-ONE FUNCTIONS
One-to-one functions are important because
they are precisely the functions that possess
inverse functions according to the following
definition.
ONE-TO-ONE FUNCTIONS
The definition states that, if f maps x
into y, then f -1 maps y back into x.
ƒ If f were not one-to-one, then f -1 would not
be uniquely defined.
f ( x) = y
for any y in B.
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ONE-TO-ONE FUNCTIONS
ONE-TO-ONE FUNCTIONS
The arrow diagram in the figure
indicates that f -1 reverses the effect of f.
Note that:
domain of f -1 = range of f
range of f -1 = domain of f
ONE-TO-ONE FUNCTIONS
ONE-TO-ONE FUNCTIONS
For example, the inverse function of
f(x) = x3 is f -1(x) = x1/3.
Do not mistake the -1 in f -1
for an exponent.
ƒ This is because, if y = x3, then
ƒ Thus, f -1(x) does not mean
Caution
1
.
f ( x)
f -1(y) = f -1(x3) = (x3)1/3 = x
1
ƒ However, the reciprocal
could be written
f ( x)
as [f(x)]-1.
ONE-TO-ONE FUNCTIONS
Example 3
If f(1) = 5, f(3) = 7, and f(8) = -10,
find f -1(7), f -1(5), and f -1(-10).
ONE-TO-ONE FUNCTIONS
Example 3
This diagram makes it clear how f -1
reverses the effect of f in this case.
ƒ From the definition of f -1, we have:
f -1(7) = 3
f -1(5) = 1
f -1(-10) = 8
because
because
because
f(3) = 7
f(1) = 5
f(8) = -10
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ONE-TO-ONE FUNCTIONS
Definition 3
The letter x is traditionally used as the
independent variable.
So, when we concentrate on f -1 rather than
on f, we usually reverse the roles of x and y
in Definition 2 and write:
f −1 ( x) = y ⇔
f ( y) = x
CANCELLATION EQUATIONS
Definition 4
By substituting for y in Definition 2 and
substituting for x in Definition 3, we get
the following cancellation equations:
f -1(f(x)) = x
for every x in A
f(f -1(x)) = x
for every x in B
CANCELLATION EQUATION 1
CANCELLATION EQUATION 2
The first cancellation equation states that,
if we start with x, apply f, and then apply
f -1, we arrive back at x, where we started.
The second equation states that
f undoes what f -1 does.
Thus, f -1 undoes what f does.
INVERSE FUNCTIONS
CANCELLATION EQUATIONS
For example, if f(x) =
x3,
then f
-1(x)
=
x1/3.
So, the cancellation equations become:
f -1(f(x)) = (x3)1/3 = x
f(f -1(x)) = (x1/3)3 = x
ƒ These equations simply states that the cube function
and the cube root function cancel each other when
applied in succession.
Now, let’s see how to compute inverse
functions.
ƒ If we have a function y = f(x) and are able to solve
this equation for x in terms of y, then, according to
Definition 2, we must have x = f -1(y).
ƒ If we want to call the independent variable x,
we then interchange x and y and arrive at
the equation y = f -1(x).
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INVERSE FUNCTIONS
Definition 5
Now, let’s see how to find the inverse
function of a one-to-one function f.
1. Write y = f(x).
2. Solve this equation for x in terms of y (if possible).
3. To express f -1 as a function of x, interchange x and y.
The resulting equation is y = f -1(x).
INVERSE FUNCTIONS
Example 4
Find the inverse function of
f(x) = x3 + 2.
ƒ By Definition 5, we first write: y = x3 + 2.
3
ƒ Then, we solve this equation for x : x = y − 2
x =
3
ƒ Finally, we interchange x and y : y =
3
ƒ So, the inverse function is: f −1 ( x) =
x−2
3
y−2
x−2
INVERSE FUNCTIONS
INVERSE FUNCTIONS
The principle of interchanging x and y
to find the inverse function also gives us
the method for obtaining the graph of f -1
from the graph of f.
However, we get the point (b, a) from
(a, b) by reflecting about the line y = x.
ƒ As f(a) = b if and only if f -1(b) = a, the point (a, b)
is on the graph of f if and only if the point (b, a) is
on the graph of f -1.
INVERSE FUNCTIONS
INVERSE FUNCTIONS
Example 5
Thus, the graph of f -1 is obtained by
reflecting the graph of f about the line
y = x.
Sketch the graphs of f ( x) = −1 − x
and its inverse function using the same
coordinate axes.
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INVERSE FUNCTIONS
Example 5
First, we sketch the curve y = −1 − x
(the top half of the parabola y2 = -1 -x,
or x = -y2 - 1).
Then, we reflect
about the line y = x
to get the graph of f -1.
INVERSE FUNCTIONS
Example 5
As a check on our graph, notice that the
expression for f -1 is f -1(x) = - x2 - 1, x ≥ 0.
ƒ So, the graph of f -1
is the right half of the
parabola y = - x2 - 1.
ƒ This seems reasonable
from the figure.
LOGARITHMIC FUNCTIONS
LOGARITHMIC FUNCTIONS
If a > 0 and a ≠ 1, the exponential function
f(x) = ax is either increasing or decreasing,
so it is one-to-one by the Horizontal Line Test.
If we use the formulation of an inverse
function given by Definition 3,
Thus, it has an inverse function f -1, which
is called the logarithmic function with base a
and is denoted by loga.
Definition 6
f −1(x) = y ⇔ f (y) = x
then we have:
log a x = y ⇔ a y = x
LOGARITHMIC FUNCTIONS
LOGARITHMIC FUNCTIONS
Thus, if x > 0, then logax is the exponent
to which the base a must be raised
to give x.
The cancellation equations, when applied to
the functions f(x) = ax and f -1(x) = logax,
become:
ƒ For example, log100.001 = - 3 because
10-3 = 0.001
log a (a x ) = x
a
log a x
=x
Definition 7
for every x ∈°
for every x > 0
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LOGARITHMIC FUNCTIONS
LOGARITHMIC FUNCTIONS
The logarithmic function loga has
domain
and (0, ∞) .
The figure shows the case where
a > 1.
ƒ Its graph is the reflection of the graph
of y = ax about the line y = x.
ƒ The most important
logarithmic functions
have base a > 1.
LOGARITHMIC FUNCTIONS
LOGARITHMIC FUNCTIONS
The fact that y = ax is a very rapidly
increasing function for x > 0 is reflected in the
fact that y = logax is a very slowly increasing
function for x > 1.
The figure shows the graphs of y = logax
with various values of the base a > 1.
LOGARITHMIC FUNCTIONS
LAWS OF LOGARITHMS
The following properties of logarithmic
functions follow from the corresponding
properties of exponential functions given
If x and y are positive numbers, then
in Section 1.5.
ƒSince loga1 = 0,
the graphs of all
logarithmic functions
pass through the point
(1, 0).
1. log a ( xy ) = log a ( x) + log a ( y )
⎛x⎞
⎟ = log a ( x) − log a ( y )
⎝ y⎠
2. log a ⎜
3. log a ( x r ) = r log a x where r is any real number
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LAWS OF LOGARITHMS
Example 6
Use the laws of logarithms to evaluate
log280 - log25.
ƒ Using Law 2, we have log 2 80 − log 2 5
NATURAL LOGARITHMS
Of all possible bases a for logarithms,
we will see in Chapter 3 that the most
convenient choice of a base is the number e,
which was defined in Section 1.5
⎛ 80 ⎞
= log 2 ⎜ ⎟
⎝ 5 ⎠
= log 2 16 = 4
because 24 = 16.
NATURAL LOGARITHM
NATURAL LOGARITHMS
The logarithm with base e is called
the natural logarithm and has a special
notation:
If we put a = e and replace loge with ‘ln’
in Definitions 6 and 7, then the defining
properties of the natural logarithm function
become:
log e x = ln x
Definitions 8 and 9
ln x = y ⇔ e y = x
ln(e x ) = x x ∈°
eln x = x
x>0
NATURAL LOGARITHMS
NATURAL LOGARITHMS
In particular, if we set x = 1,
we get:
Find x if ln x = 5.
ln e = 1
E. g. 7—Solution 1
ƒ From Definition 8, we see that ln x = 5
means e5 = x.
ƒ Therefore, x = e5.
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NATURAL LOGARITHMS
E. g. 7—Solution 1
If you have trouble working with the ‘ln’
notation, just replace it by loge.
ƒThen, the equation becomes loge x = 5.
ƒSo, by the definition of logarithm, e5 = x.
NATURAL LOGARITHMS
E. g. 7—Solution 2
Start with the equation ln x = 5.
Then, apply the exponential function to both
sides of the equation: eln x = e5
ƒ However, the second cancellation equation
in Definition 9 states that eln x = x.
ƒ Therefore, x = e5.
NATURAL LOGARITHMS
Solve the equation
Example 8
e5 - 3x =
10.
ƒ We take natural logarithms of both sides of the
equation and use Definition 9:
5−3 x
ln(e ) = ln10
5 − 3 x = ln10
3 x = 5 − ln10
1
x = (5 − ln10)
3
ƒ As the natural logarithm is found on scientific
calculators, we can approximate the solution—
to four decimal places: x ≈ 0.8991
NATURAL LOGARITHMS
Express ln a +
logarithm.
1
2
Example 9
ln b as a single
ƒ Using Laws 3 and 1 of logarithms, we have:
ln a + 12 ln b = ln a + ln b1/ 2
= ln a + ln b
= ln(a b )
NATURAL LOGARITHMS
CHANGE OF BASE FORMULA
Formula 10
The following formula shows that
logarithms with any base can be
expressed in terms of the natural
logarithm.
For any positive number a (a ≠ 1),
we have:
ln x
log a x =
ln a
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CHANGE OF BASE FORMULA
Proof
Let y = logax.
ƒ Then, from Definition 6, we have ay = x.
ƒ Taking natural logarithms of both sides
of this equation, we get y ln a = ln x.
ƒ Therefore, y =
ln x
ln a
NATURAL LOGARITHMS
Scientific calculators have a key for
natural logarithms.
ƒ So, Formula 10 enables us to use a calculator
to compute a logarithm with any base—as shown
in the following example.
ƒ Similarly, Formula 10 allows us to graph any
logarithmic function on a graphing calculator
or computer.
Example 10
Evaluate log8 5 correct to six
decimal places.
ƒ Formula 10 gives: log 8 5 =
NATURAL LOGARITHMS
ln 5
≈ 0.773976
ln 8
NATURAL LOGARITHMS
The graphs of the exponential function y = ex
and its inverse function, the natural logarithm
function, are shown.
ƒ As the curve y = ex
crosses the y-axis with
a slope of 1, it follows
that the reflected curve
y = ln x crosses the
x-axis with a slope of 1.
NATURAL LOGARITHMS
NATURAL LOGARITHMS
Example 11
In common with all other logarithmic functions
with base greater than 1, the natural
logarithm is an increasing function defined on
(0, ∞ ) and the y-axis is a vertical asymptote.
Sketch the graph of the function
y = ln(x - 2) -1.
ƒ We start with the graph of y = ln x.
ƒ This means that the values of ln x become
very large negative as x approaches 0.
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NATURAL LOGARITHMS
Example 11
NATURAL LOGARITHMS
Example 11
ƒ Using the transformations of Section 1.3,
we shift it 2 units to the right—to get the
graph of y = ln(x - 2).
ƒ Then, we shift it 1 unit downward—to
get the graph of y = ln(x - 2) -1.
NATURAL LOGARITHMS
NATURAL LOGARITHMS
Although ln x is an increasing function,
it grows very slowly when x > 1.
To illustrate this fact, we compare
approximate values of the functions
y = ln x and y = x½ = x in the table.
ƒ In fact, ln x grows more slowly than
any positive power of x.
NATURAL LOGARITHMS
INVERSE TRIGONOMETRIC FUNCTIONS
We graph the functions here.
When we try to find the inverse
trigonometric functions, we have
a slight difficulty.
ƒ Initially, the graphs grow at comparable rates.
ƒ Eventually, though, the root function far surpasses
the logarithm.
ƒ As the trigonometric functions are not
one-to-one, they don’t have inverse functions.
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INVERSE TRIGONOMETRIC FUNCTIONS
INVERSE TRIGONOMETRIC FUNCTIONS
Here, you can see that the sine function
y = sin x is not one-to-one.
However, here, you can see that
the function f(x) = sin x, − π 2 ≤ x ≤ π 2 ,
is one-to-one.
ƒ Use the Horizontal Line Test.
INVERSE SINE FUNCTION / ARCSINE FUNCTION
INVERSE SINE FUNCTIONS
The inverse function of this restricted sine
function f exists and is denoted by sin-1 or
As the definition of an inverse function states
that
f −1 ( x ) = y ⇔ f ( y ) = x
arcsin.
ƒ It is called the inverse sine function or the arcsine
function.
we have:
sin −1 x = y ⇔ sin y = x and −
π
2
≤ y≤
ƒ Thus, if -1 ≤ x ≤ 1, sin-1x is the number between
and π 2 whose sine is x.
INVERSE SINE FUNCTIONS
Evaluate:
⎛1⎞
a. sin −1 ⎜ ⎟
⎝2⎠
1
b. tan(arcsin )
3
Example 12
INVERSE SINE FUNCTIONS
We have
π
2
−π 2
Example 12 a
⎛1⎞ π
sin −1 ⎜ ⎟ =
⎝2⎠ 6
ƒ This is because sin (π / 6 ) = 1/ 2 , and π / 6 lies
between −π / 2 and π / 2 .
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INVERSE SINE FUNCTIONS
Example 12 b
1
1
Let θ = arcsin , so sin θ = .
3
3
INVERSE SINE FUNCTIONS
Example 12 b
ƒ This enables us to read from
the triangle that:
ƒ Then, we can draw a right triangle with angle θ.
1
1
tan(arcsin ) = tan θ =
3
2 2
ƒ So, we deduce from the Pythagorean Theorem
that the third side has length 9 − 1 = 2 2 .
INVERSE SINE FUNCTIONS
INVERSE SINE FUNCTIONS
In this case, the cancellation equations
for inverse functions become:
The inverse sine function, sin-1, has
domain [-1, 1] and range [ −π / 2, π / 2] .
sin −1 (sin x) = x
−1
sin(sin x) = x
for −
π
≤x≤
π
Its graph is shown.
2
2
for − 1 ≤ x ≤ 1
INVERSE SINE FUNCTIONS
INVERSE COSINE FUNCTIONS
The graph is obtained from that of
the restricted sine function by reflection
about the line y = x.
The inverse cosine function is handled
similarly.
ƒ The restricted cosine function f(x) = cos x, 0 ≤ x ≤ π,
is one-to-one.
ƒ So, it has an inverse function denoted by cos-1 or arccos.
cos −1 x = y ⇔ cos y = x and 0 ≤ y ≤ π
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INVERSE COSINE FUNCTIONS
INVERSE COSINE FUNCTIONS
The cancellation equations are:
The inverse cosine function,cos-1, has
domain [-1, 1] and range [0, π ] .
cos −1 (cos x) = x for 0 ≤ x ≤ π
Its graph is shown.
cos(cos −1 x) = x for − 1 ≤ x ≤ 1
INVERSE TANGENT FUNCTIONS
INVERSE TANGENT FUNCTIONS
The tangent function can be made
one-to-one by restricting it to the interval
( −π / 2, π / 2 ) .
Thus, the inverse tangent
function is defined as
the inverse of the function
f(x) = tan x,
−π / 2 < x < π / 2 .
ƒ It is denoted by tan-1
or arctan.
tan −1 x = y ⇔ tan y = x and −
INVERSE TANGENT FUNCTIONS
E. g. 13—Solution 1
Simplify the expression cos(tan-1x).
INVERSE TANGENT FUNCTIONS
π
2
< y<
π
2
E. g. 13—Solution 1
ƒ Therefore,
sec 2 y = 1 + tan 2 y = 1 + x 2
ƒ Let y = tan-1x.
ƒ Then, tan y = x and −π / 2 < y < π / 2 .
ƒ We want to find cos y.
ƒ However, since tan y is known, it is easier
to find sec y first.
sec y = 1 + x 2
Since sec y > 0 for − π / 2 < y < π / 2
ƒ Thus,
cos(tan −1 x) = cos y =
1
1
=
sec y
1 + x2
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INVERSE TANGENT FUNCTIONS
E. g. 13—Solution 2
Instead of using trigonometric identities,
it is perhaps easier to use a diagram.
ƒ If y = tan-1x, then tan y = x.
ƒ We can read from the figure (which illustrates
the case y > 0) that:
cos(tan −1 ) = cos y =
INVERSE TANGENT FUNCTIONS
The inverse tangent function, tan-1 = arctan,
has domain and range (−π / 2, π / 2).
Its graph is shown.
1
1 + x2
INVERSE TANGENT FUNCTIONS
INVERSE FUNCTIONS
We know that the lines x = ±π / 2 are
vertical asymptotes of the graph of tan .
The remaining inverse trigonometric
functions are not used as frequently and
are summarized here.
ƒ The graph of tan-1 is obtained by reflecting the graph
of the restricted tangent function about the line y = x.
ƒ It follows that the lines
y = π/2 and y = -π/2
are horizontal
asymptotes of the
graph of tan-1.
Definition 11
(
(
y = csc−1 x(| x |≥1) ⇔ csc y = x and y ∈ 0,π / 2⎤⎦ ∪ π ,3π / 2⎤⎦
)
y = sec−1 x(| x |≥1) ⇔ sec y = x and y ∈⎡⎣0,π / 2 ∪ ⎡⎣π ,3π / 2
)
y = cot −1 x(x ∈° ) ⇔ cot y = x and y ∈(0,π )
INVERSE FUNCTIONS
INVERSE FUNCTIONS
The choice of intervals for y in
the definitions of csc-1 and sec-1 is
not universally agreed upon.
For instance, some authors use
y ∈ [ 0, π / 2 ) ∪ (π / 2, π ] in the definition
of sec-1.
ƒ You can see from the
graph of the secant
function that both this
choice and the one in
Definition 11 will work.
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