55

Solving Radical Equations (Section 3.4)

Example. Simplify the following.

p64 3

p27 3

p�8

4p81 5

p�32 4

p1

How do you use your calculator to help with things like 4p15?

Example. Use your calculator to round the following to three decimal places.

3p�83 4

p76 5

p19

56

How do we solve an equation like 3p2� x = 4?

Example. Solve each of the following equations.p4x+ 1 = 3

py � 1 + 4 = 0

5p3x+ 4 = 2 3

p6x� 9 + 8 = 5

57

How do we solve an equation likepx+ 1 + 1 = x?

Example. Solve each of the following equations.p7x+ 4 = x+ 2

px� 3 + 5 = x

58

Graphs of Rational Functions (Section 4.5)

Definition. A rational function if a function of the form:

where p(x) and q(x) are polynomials and q(x) 6= 0.

Example. Graph the following functions on your graphing calculator and determine the domain.

f(x) =3

x+ 5f(x) =

x+ 1

x

2 � x� 6

Remark. The domain for a rational function f(x) =p(x)

q(x)is:

Example. Determine the domain for each of the following functions.

f(x) =2x� 11

x

2 + 2x� 8f(x) =

x

2 � 4x

x

3 � x

59

Graph f(x) =x

2 � 4x

x

3 � x

on your graphing calculator:

�10�9�8�7�6�5�4�3�2�1 1 2 3 4 5 6 7 8 9 10

�10

�9

�8

�7

�6

�5

�4

�3

�2

�1

1

2

3

4

5

6

7

8

9

10

Remark. To find vertical asymptotes for a rational function f(x) =p(x)

q(x):

(1) Reduce f(x) by canceling out any common factors between p(x) and q(x).(2) Set the new denominator equal to zero.(3) The vertical asymptotes correspond to the answers found above.

Example. Determine the vertical asymptotes for each of the following rational functions.

f(x) =x

2 � 2x

x

2 + 2x� 8g(x) =

2x+ 6

x

2 + 7x� 8

60

Remark. To find horizontal asymptotes for a rational function f(x) =p(x)

q(x), you only look at

the leading terms of p(x) and q(x). Consider f(x) =ax

n + · · ·bx

m + · · · . If:

(1) n > m, then there is no horizontal asymptote.(2) n < m, then the horizontal asymptote is y = 0

(3) n = m, then the horizontal asymptote is y =a

b

.

Example. Determine the horizontal asymptotes for each of the following rational functions.

f(x) =x

2 � 2x

x

2 + 2x� 8g(x) =

2x+ 6

x

2 + 7x� 8h(x) =

2x3 + 6

x

2 � 1

**Graphs can cross horizontal asymptotes ... but graphs will NEVER cross verticalasymptotes

Remark. Sometimes there is an asymptote that is neither horizontal nor vertical. This canoccur when there is no horizontal asymptote. There will be an oblique asymptote (also called a

slant asymptote for the function f(x) =p(x)

q(x)when deg p(x) = deg q(x) + 1. To find an oblique

asymptote:

(1) Perform long division (or synthetic division, if the division looks like x+ c or x� c)(2) The quotient is the oblique asymptote

Example. Determine the oblique asymptote for each of the following rational functions.

f(x) =x

2 � 3x+ 4

x+ 3g(x) =

2x3

x

2 + 1

61

Solving Rational Equations (Section 3.4)

How do we solve an equation likex� 8

3+

x� 3

2= 0?

. . . or how would you solve an equation like2x

x+ 7=

5

x� 3?

62

Example. Solve the following equations.

x+6

x

= 51

2+

2

x

=1

3+

3

x

2

x� 1=

3

x+ 2

3y + 5

y

2 + 5y+

y + 4

y + 5=

y + 1

y

63

More About Graphs of Rational Functions (Section 4.5)

Previously, we learned how to find the domain, vertical asymptotes, and horizontal asymptotesfor a rational function. This information told us some keys as to what the graph of the functionwould look like. Now, we review how to find x- and y-intercepts. To graph a rational functionaccurately, you would need to do all of these things and then to create a table to find additionalpoints.

Example. Let’s find the important parts of the graphs of these rational functions.

f(x) =x

2 + 3x� 10

x

2 � 2xg(x) =

x

2 � 4x

x

3 � x

�10�9�8�7�6�5�4�3�2�1 1 2 3 4 5 6 7 8 9 10

�10

�9

�8

�7

�6

�5

�4

�3

�2

�1

1

2

3

4

5

6

7

8

9

10

�10�9�8�7�6�5�4�3�2�1 1 2 3 4 5 6 7 8 9 10

�10

�9

�8

�7

�6

�5

�4

�3

�2

�1

1

2

3

4

5

6

7

8

9

10

64

Word problems with Rational Functions

Example. A basketball player has made 5 of the last 7 free throws. How many more consecutivefree throws do they need to make to have an average of 80% made free throws?

Example. The function below gives the concentration of the saline solution after adding x

milliliters of a 0.5% solution to 100 milliliters of a 2% solution.

f(x) =100(0.02) + x(0.005)

100 + x

How many milliliters of the 0.5% solution must be added to have a combined concentration of0.9% solution?

Example. The percentage of a drug in a person’s bloodstream t hours after its injection isapproximated by the function:

P (t) =5t

4t2 + 5Determine the percentage of the drug in a person’s bloodstream 5.5 hours after the drug isinjected.

65

Compositions of Functions (Section 2.3)

In the beginning of the semester, we did problems like this one:

For f(x) = 3x� 1 and g(x) = x

2 + 4x� 1 find the following:

f(4 + h) g(1 + h)

This leads into the composition of functions. For the two functions f(x) and g(x), the compo-sition functions f � g and g � f are defined as:

Example. Consider the functions f(x) = 3x+ 1 and g(x) = x

2 + 4. Find:(f � g)(x) (g � f)(x)

Remark. Notice that:

66

Example. Let f(x) = 1� 2x, g(x) =px+ 1, and h(x) = x

2 + 8. Find:(g � f)(x) (f � f)(x)

(f � h)(x) (h � g)(x)

It is very important in Calculus that you can do compositions of functions, because the derivativeis defined using compositions of functions via the di↵erence quotient. For a function f(x), thedi↵erence quotient of f(x) is:

f(x+ h)� f(x)

h

Let’s find the di↵erence quotient for f(x) = 4x� 1.

67

Example. Find the di↵erence quotient for each function.f(x) = x

2 + 3 g(x) = 2x2 � x

Example. Consider f(x) = 3x� 2 and g(x) =x+ 2

3. Find (f � g)(x) and (g � f)(x)

68

Inverses Functions (Section 5.1)

In the previous example, f(x) = 3x� 2 and g(x) =x+ 2

3are inverses:

Definition. Two functions f and g are called inverses if

Example. Determine if the given two functions are inverses of each other.

f(x) = 3px+ 1 and g(x) = x

3 + 1 f(x) =1� x

x

and g(x) =1

x+ 1

69

So when does a function have an inverse? And then how do we find the inverse?

Example. Determine the inverse of f(x) (if it exists).

f(x) = x

2 � 4x+ 1 f(x) =x+ 5

7

70

Example. Determine the inverse of the given function (if it exists).

f(x) =3

x� 1g(x) = 5

px+ 2

We now want to look at two di↵erent types of functions, called logarithmic functions and expo-

nential functions. It will turn out that these two types of functions are inverses of each other,i.e. the inverse of a logarithmic function will be an exponential function and the inverse of anexponential function will be a logarithmic function.

Exponential Functions (Section 5.2)

Definition. The function

x is a real number, a > 0, and a 6= 1, is called the exponential function with base a.

Example. Consider the exponential functions

f(x) = 2x and g(x) =

✓1

2

◆x

and h(x) = e

x

.

Find the following output values.

f(�2) = g(�2) = h(�2) =

f(0) = g(0) = h(0) =

f(2) = g(2) = h(2) =

71

Let’s draw the graphs of f(x) = 2x and f(x) =

✓1

2

◆x

:

�10�9�8�7�6�5�4�3�2�1 1 2 3 4 5 6 7 8 9 10

�10

�9

�8

�7

�6

�5

�4

�3

�2

�1

1

2

3

4

5

6

7

8

9

10

�10�9�8�7�6�5�4�3�2�1 1 2 3 4 5 6 7 8 9 10

�10

�9

�8

�7

�6

�5

�4

�3

�2

�1

1

2

3

4

5

6

7

8

9

10

Now, let’s draw the graphs of f(x) = 3x and f(x) =

✓1

3

◆x

:

�10�9�8�7�6�5�4�3�2�1 1 2 3 4 5 6 7 8 9 10

�10

�9

�8

�7

�6

�5

�4

�3

�2

�1

1

2

3

4

5

6

7

8

9

10

�10�9�8�7�6�5�4�3�2�1 1 2 3 4 5 6 7 8 9 10

�10

�9

�8

�7

�6

�5

�4

�3

�2

�1

1

2

3

4

5

6

7

8

9

10

So graphs of exponential functions look like:

�10�9�8�7�6�5�4�3�2�1 1 2 3 4 5 6 7 8 9 10

�10

�9

�8

�7

�6

�5

�4

�3

�2

�1

1

2

3

4

5

6

7

8

9

10

72

Application to Compound Interest (Section 5.2)

If P dollars are invested at an interest rate r (written as a decimal), compounded n times peryear, then the amount A of money in the account after t years is given by:

A = P

⇣1 +

r

n

⌘nt

Example. If $500 is invested at a rate of 1.25%, compounded monthly, how much money willbe in the account after 5 years?

Example. If $1200 is invested at a rate of 2.5%, compounded semiannually, how much moneywill be in the account after 2 years?

73

As the number of times you compound interest per year increases, the amount in the accountafter t years also increases. Consider investing $100 at a rate of 2% for a time period of 5 years.So

A = 100

✓1 +

.02

n

◆5n

If interest is compounded continuously, then the amount A in the account after t years is:

A = Pe

rt

Example. If $500 is invested at a rate of 1.25%, compounded continuously, how much moneywill be in the account after 5 years?

Example. If $1200 is invested at a rate of 2.5%, compounded continuously, how much moneywill be in the account after 2 years?

74

Logarithms (Section 5.3)

What does loga

x mean????

Example. Calculate the following.log

2

8 log5

1 log1

00.001

log3

81 log2

✓1

8

◆log

4

16

Definition. The function

is the logarithmic function having base a, where a > 0 is a real number.

Two bases that occur often have special notations:

75

Example. Compute the following. Round your answers to two decimal places.log 7 ln 2 log 30 ln 20

Your calculator has buttons for ln x and log x, but how do you calculate other base logarithms?

Example. Compute the following. Round your answers to two decimal places.log

3

15 log2

6 log6

30

You can convert between logarithmic and exponential equations using the following:

76

Example. Convert the following logarithmic equations to exponential equations.log

5

5 = 1 log 7 = 0.845 ln 40 = x

Example. Convert the following exponential equations to logarithmic equations.

e

3 = t 5�3 =1

125103 = 1000

Graphs of Inverse Functions (Section 5.1)

Let’s compare the graphs of two inverse functions. Graph f(x) = 2x � 3 and f

�1(x) =x+ 3

2on the same coordinate axes:

�10�9�8�7�6�5�4�3�2�1 1 2 3 4 5 6 7 8 9 10

�10

�9

�8

�7

�6

�5

�4

�3

�2

�1

1

2

3

4

5

6

7

8

9

10

77

Let’s try another pair of inverse functions. Graph f(x) = x

3 + 2 and f

�1(x) = 3px� 2 on the

same coordinate axes:

�10�9�8�7�6�5�4�3�2�1 1 2 3 4 5 6 7 8 9 10

�10

�9

�8

�7

�6

�5

�4

�3

�2

�1

1

2

3

4

5

6

7

8

9

10

Since f(x) = a

x and f

�1(x) = loga

(x), we can use the graph of f(x) = a

x to graph

f

�1(x) = loga

(x).

Example. Sketch the graph of each logarithmic equation.y = log

2

x y = log 12x

�10�9�8�7�6�5�4�3�2�1 1 2 3 4 5 6 7 8 9 10

�10

�9

�8

�7

�6

�5

�4

�3

�2

�1

1

2

3

4

5

6

7

8

9

10

�10�9�8�7�6�5�4�3�2�1 1 2 3 4 5 6 7 8 9 10

�10

�9

�8

�7

�6

�5

�4

�3

�2

�1

1

2

3

4

5

6

7

8

9

10

78

Solving Exponential and Logarithmic Equations (Section 5.5)

Logarithm and exponential functions are inverses, so:

loga

(ax) = x and a

loga(x) = x

Example. Solve the following equations:2x+3 = 5 log

5

(3x� 1) = 12

54x�7 = 125 log5

(8� 7x) = 3

2x = 40 ln x = �2

79

Properties of logarithms:

• loga

M + loga

N = loga

MN

• loga

M � loga

N = loga

M

N

• p loga

M = loga

M

p

So, how do we solve logarithmic equations like

log8

(x+ 1)� log8

x = 2

Example. Solve the following equations:log

2

(x+ 1) + log2

(x� 1) = 3 log x+ log(x+ 4) = log 12

80

One more example . . . solve

ln(x+ 8) + ln(x� 1) = 2 ln x

Let’s take a look back at solving exponential equations.

Example. Solve the following equations:

1000e0.09t = 5000 250� (1.87)x = 0

81

Example. Suppose that $1,500 is invested at an interest rate of 1.75%, compounded quarterly.How long would it take for the amount of money in the account to become $2,000?

Example. Suppose that $5,000 is invested at an interest rate of 5.4%, compounded continuously.How long would it take for the amount of money to double?