1
Math 101 Meeting 12
Inverse Functions & Logarithms
Part 1 Inverse Functions
Motivating Example: Solve the equation
for x.
Multiply both sides of the equation by 5.
Subtract 4 from both sides of the equation.
Divide both sides of the equation by 3.
The equation
function.
is the equation for a linear function. Make a flow diagram for this
1. Multiply by 3.
y
2. Add 4.
3. Divide by 5.
4.
When you solved for x above, you arrived at the equation
. This equation is also the
equation for a function. For this function, however, y represents the input (independent variable)
and x represents the output (dependent variable). Make a flow diagram for this function.
x
1. Multiply by 5.
2. Subtract 4.
3. Divide by 3.
y
x
How do the flow diagrams of these two functions compare? (Look at both the mathematical
operations used AND the order in which operations are performed.)
Notice that the two functions use opposite operations. For example, we multiply by 3 for the first
function and we divide by 3 for the second. We add 4 for the first function and we subtract 4 for
the second. We divide by 5 for the first function and we multiply by 5 for the second function.
Also, notice that these opposite operations are performed in reverse order. For example, step 1
for the first function is to multiply by 3, while division by 3 is the final step for the second
function. Dividing by 5 is the final step for the first function, while multiplying by 5 is the first
step for the second function.
Complete the following input/output tables for these two functions.
x
2
7
12
2
5
8
y
2
5
8
2
7
12
The two functions in the last example are related to each other. In fact, they are said to be
inverses of each other. Why do you think they might be called inverses?
When you compare the flow diagrams for the two functions, you can see that the two functions
describe inverse processes. One function effectively “undoes” what the other function “does”.
The result is that that the input and output variables for these functions are reversed. Using
function notation, we could write this as follows:
If two functions f and g are inverses of each other and
, then
.
Graphically, this would mean that if (x, y) is a point on the graph of a function, then (y, x) would
be a point on the graph of the inverse function.
Example 1: For each of the following functions, find the inverse function and demonstrate that
the two functions are inverses of each other.
(a)
We determine the inverse function by solving the above equation for x:
Add 4 to both sides of the equation.
Divide both sides of equation by 2.
To demonstrate that
and
are inverse functions, let’s choose a value
for x, say x = 6. Then
. Now, note that, if y = 8, then
. So, we’re back to the original value that we chose for x.
(b)
We determine the inverse function by solving the above equation for t:
Subtract 12 from both sides of the equation.
Divide both sides of equation by 3.
To demonstrate that
and
are inverse functions, let’s choose a value
for t, say t = -3. Then
. Now, note that, if p = 3, then
. So, we’re back to the original value that we chose for t.
Example 2: Are
and
inverses of each other?
If we solve the equation
inverses of each other:
for u, we should be able to see if these functions are
Subtract 4 from both sides of the equation.
Divide both sides of equation by -3.
Since the functions
can not be inverses.
and
are not the same function, the above functions
3
Example 3: Suppose a population is growing linearly by 1.1 million people per year. Currently,
the population is at 250 million.
(a) Let p represent the size of the population t years after the current year. Determine the
equation of the function that describes this growth.
The equation of the function is
.
(b) Solve the equation in part (a) for t. In other words, find the inverse function.
Subtract 250 from both sides of the equation.
Divide both sides of equation by 1.1.
(c) How large is the population after 15 years?
We have to determine the value of p when t = 15. We see that
million people.
(d) When will the population reach 300 million? Use the inverse function from part (b) to
answer this!!
Now, we have to determine the value of t when p = 300. Using the inverse function from
part (b), we see that
years.
Part 2 Logarithms
In all of the examples in the previous section, we obtained inverses of linear functions. Go back
and look at these inverse functions. What type of functions were the inverses? Were they linear,
exponential or something else?
The inverses of linear functions are themselves linear functions.
So, what about exponential functions? If you try to find the inverse of an exponential function,
what kind of function will you obtain?
Motivating Example: Suppose a population decays by 2.3% every 6 years. The current size of
the population is 800.
(a) Let p represent the size of the population t years after the current year. Determine the
equation of the function that describes this growth.
The population is growing exponentially. So, the equation of the function is
.
(b) Solve
for t. In other words, find the inverse function.
Our goal is to isolate t on one side of the equation. We can divide both side of this
equation by 800 which gives
.
We are stuck! The variable we are trying to solve for is in an exponent. Up to this point,
we have not learned anything that will help us solve an equation like this. We will need
something entirely new. This means that the inverse of this exponential function will be a
type of function that we have not encountered yet—a logarithmic function. We will
return to this example, but first we need to learn more about logarithmic functions.
There are many different logarithmic functions. Shown below are equations for four different
logarithmic functions.
y  log 3 x
y  log 5.6 x
y  log 0.94 x
y  log 11 x
The letter x represents the input variable and y represents the output variable. In each example,
the ‘logc’ represents the function name. In short, for each positive number c,
is a
different function. The number c is called the ‘base’ and ‘
’ is referred to as a ‘base-c
logarithm’.
What’s special about logarithmic functions?
The functions
and
are inverses of each other.
To see what this really means, complete the input/output tables below. For the first table,
determine the outputs by hand. For the second table, use the LOG button on your calculator
(short for ‘log10’) to determine the outputs.
x
-2
0.01
x
-1
0.1
0
1
1
10
2
100
0.01
-2
0.1
-1
1
0
10
1
100
2
What do you notice?
The input and output values of these two functions are reversed. For example, (-2, 0.01)
is an input/output pair for
while (0.01, -2) is an input/output pair for
𝑦=
.
Example 4: Let’s see if you have the idea clear in your mind. Complete the following tables
without using a calculator. Start with the table on the left!
x
-3
1/8
-2
1/4
-1
1/2
0
1
1
2
2
4
3
8
x
1/8
-3
1/4
-2
1/2
-1
1
0
2
1
4
2
8
3
5
Because exponential functions and logarithmic functions are inverses of each other, the
following equations are equivalent:
and
The first equation says that (p, q) is an input/output pair for the function
equation says that (q, p) is an input/output pair for the function
.
. The second
Example 5: For each of the following equations, write an equivalent equation using logarithms.
(a)
(b)
(c)
Example 6: For each of the following equations, write an equivalent equation using exponents.
(a)
(b)
(c)
Example 7: Solve each of the following equations.
(a)
(b)
(c)
Divide both sides of equation by 8.
(d)
Divide both sides of equation by 23.
Multiply both sides of the equation
by 4.
Let’s use the graphing calculator to determine decimal values for each of the solutions in the last
example. Recall that your calculator has an ‘LOG’ button which enables you to determine
decimal values for base-10 logarithms (e.g.
). If you look at the solutions above,
however, none of them is written as a base-10 logarithm. So, what do we do?
What we need to do is convert the logarithms from the last example into base-10 logarithms. To
do this, you will need to use a property of logarithms which you will learn if you take college
algebra or pre-calculus:
(Change-of-Base Property)
This property converts a base-c logarithm into a base-d logarithm. Although d can be an positive
number, we will take d = 10 since we want to convert to base-10. To illustrate, suppose you want
to find the decimal value of
. We rewrite this as follows:
Now, we enter “LOG(22) / LOG(4)” into the calculator, and we find that
So, go ahead and determine the decimal values for the solutions in the last example:
(a)
(b)
(c)
(d)
Note: It is standard practice to write ‘
’ in place of ‘
’. Base-10 logarithms are used
extensively in the sciences, and are generally referred to as the common logarithm.
Example 8: A substance decays 7% every 6 hours. Initially, there were 70 grams.
(a) Determine an equation for the function that describes this decay.
Let y represent the amount of substance remaining after x hours since the present hour.
Then, the equation is
.
(b) How long will it take until half of the initial amount is left?
We want to know the value of x when y = 35. In other words, we need to solve the
equation
for x.
Divide both side of equation by 70.
7
Multiply both side of equation by 6.
Let’s determine a decimal value for this number:
So, it will take approximately 57 hours for the substance to reduce to half of its original
amount.
Motivating Example, continued: Recall that we were trying to determine the inverse of the
function
where t represented the number of years since the current year and p
represented the size of a population. Let’s try this again. We need to solve the equation
for t.
Divide both sides of equation by 800.
This equation represents the inverse function.
Now, use the inverse function to determine how long it will take before the population drops to
half of its original size?
Since the initial population size was 800, we want to find the value of t when p = 400.
The value is obtained by substituting p = 400 into the equation for the inverse:
So, it will take approximately 179 years for the population to decay to half of its original
size.
Math 101
Meeting 12 Homework Assignment
1. (a) Flow diagrams for two functions are given below. Draw flow diagrams for their
inverses.
Function 1:
x
1. Add 5.
2. Multiply by 2.
y
1. Divide by 3.
2. Add 1.
y
Function 2:
x
(b) Determine equations for the two functions in part (a), and their inverses.
2. Determine inverse functions for each of the following linear functions.
(a)
(b)
(c)
(d)
(e)
(f)
3. Use the inverse functions that you obtained in exercise 2 to answer each of the following
parts.
(a) In exercise 2 (b), determine the value of x for which y = 8.
(b) In exercise 2 (e), determine the value of t for which v = 5.
(c) In exercise 2 (f), determine the value of a for which g = 1.
4. The functions
and
are inverses of each other. Determine which
function would be easiest to use to determine…
(a) the value of r when y = 3.
(b) the value of y when r = -2.
5. An input/output table for a function is shown below. Use it to make an input/output table
for its inverse.
Input
Output
-3
15
2
7
5
3
7
-3
10
-7
6. The points (3, 1), (4, 3), (7, 5) and (10, 11) lie on the graph of a function. Determine four
points that will lie on the graph of its inverse.
9
7. For each of the following exponential and logarithmic functions, determine which letter
represents the input variable and which letter represents the output variable for that
function. Then, determine the inverse functions and determine which letter represents the
input variable and which letter represents the output variable for that function.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
8. Use your graphing calculator to determine decimal values for each of the following.
(a)
(b)
9. Solve each of the following equations. Leave your answer given in terms of logarithms.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
10. Let P represent the size of a population after t years. Assume that the population is
currently at 23,000 and that it is growing by 2.1% every 10 years.
(a) Determine an equation for the function that describes this growth over time.
(b) Determine the inverse of the function in part (a).
(c) Determine the size of the population after 35 years.
(d) Determine how long it will take before the population triples. (Which function, the
one you found in part (a) or the one you found in part (b), will most easily answer this
question.
11. Recall from Mini-Project 2 that the U.S. energy footprint in 2003 was estimated at 4.515
billion global acres.
(a) If the U.S. adopted policies that would force the U.S. energy footprint to be cut by 5%
every 5 years, create a function that would describe this decay.
(b) Determine how long it will take to reduce the footprint by 80%? Answer this by
solving an equation.
(c) Suppose we want to cut the footprint by 80% by the year 2050, by cutting the
footprint by 5% every n years. Determine the value of n by solving an equation.
Math 101
Meeting 12 Homework Solutions
1. (a) The flow diagram for Function 1’s inverse is:
1. Divide by 2.
2. Subtract 5.
y
(b) The equation for Function 1 is
1 is
.
x
The equation for the inverse of Function
2. (b)
(e)
3. (a) When y = 8,
.
5.
Input
Output
6.
15
-3
7
2
3
5
-3
7
-7
10
(1, 3), (3, 4), (5, 7) and (11, 10) will lie on the graph of the inverse.
7. (a) For the function
, t represents the input variable and y represents the output
variable. For the inverse function
, y represents the input variable and t
represents the output variable.
8. (a)
9. (a)
(c)
(e) Since
(g)
,
.
10. (a)
(b)
(c) When t = 35 years,
(d) When P = 3(23,000),
people.
years.