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Hartfield – College Algebra (Version 2015b - © Thomas Hartfield) Unit FIVE | of 29

Topic 33: One-to-One Functions Definition: A function f is said to be one-to-one

if for every value f(x) in the range of f there is exactly one corresponding x-value in the domain of f.

Ex. Are the following functions one-to-one over their domains?

2

3

3

2

2

2

2

f x x

g x x

h x x

H x x x


The Horizontal Line Test A function f is one-to-one if every horizontal line applied to the graph of f intersects f at most one. Are the graphs below representing the graphs of a function? Are they representing the graphs of a one-to-one function?

Any function which is not one-to-one over its entire domain can be amended to be one-to-one with appropriate restrictions on its domain.

Consider: 2 2g x x


Topic 34: Inverse Functions (Part 1) An inverse operation is an operation which reverses an original operation. Subtraction in the inverse operation of addition and addition is the inverse operation of subtraction. Inverses are possible no matter what the original value is: Ex. 1a Starting from x, find the inverse

operation to x + 2. Ex. 1b Starting from x, find the inverse

operation to 4x. With respect to roots, odd powers and odd indexed roots are always inverse operations. Even powers and even indexed roots can be inverse operations but only when settling on the domain of x and the range of the output.

Definition: An inverse function is a one-to-one function such that if a one-to-one function f with domain A and range B exists, then the inverse function f −1 (read as “f inverse”) has domain B and range A and the following property is satisfied:

1f x y f y x for all x in A and all y in B. Ex 2 Given that f(2) = 7, what is f -1(7)?


We can prove that two functions are inverses of each other by using the Property of Inverse Functions. Property of Inverse Functions For two functions f and g to be inverse functions of each other, the following conditions must be established:

1. f and g must be one-to-one functions over their given domains,

2. the domain of f must be the same as the range of g and the domain of g must be the same as the range of f, and

3. ( ) ( ) .f g x x g f x

Ex. 3 Determine if functions f and g are inverse functions of each other.

12 12

xf x x g x


Ex. 4 Determine if functions f and g are inverse functions of each other.

21 1, 0f x x g x x x

Graphs of Inverse Functions Since inverse functions effectively reverse the roles of the independent and dependent variables, the graph of an inverse function should satisfy the following pair of properties:

1. For every point (a, b) on the graph of f, the graph of f −1 should include a point (b, a).

2. The graph of f −1 should demonstrate symmetry to f with respect to the line y = x.


Ex. 5 Find the graph of the inverse of the function graphed below. Then state the domain and range of both f and f −1.

Ex. 6 Find the graph of the inverse of the function graphed below. Then state the domain and range of both f and f −1.


Topic 35: Inverse Functions (Part 2) To find the inverse function of a given function f:

1. Determine if f is one-to-one and its domain and range.

2. Apply one of the methods for finding the rule of

an inverse function. 3. Resolve any issues with the rule to ensure that

f −1 is one-to-one and has the appropriate domain and range.

To find the rule of an inverse function, you may apply one of two methods, each with its advantages and drawbacks. Verbal method: Write the steps of the function

that are applied to the variable. The inverse function’s rule will apply the inverse operations in the reverse order.

Symbolic method: Define the variable of the

function as y. Invert the roles of the independent variable (usually written as x) and y. Solve for y which will represent the rule of the inverse function.


Ex. 1 Find the inverse function of f. 3 4f x x

Ex. 2 Find the inverse function of f. 2 2 4f x x


Ex. 3 Find the inverse function of f. 21 3, 1f x x x

Ex. 4 Find the inverse function of f. 2 5f x x


Topic 36: Exponential Functions Definition: The exponential function with base a

is defined by , 0.xf x a a

For a ≠ 1, the domain is , and the range is 0, . Note that a power function is defined to be a function with a variable base x raised to a fixed exponent. When a power function has an integer exponent, you get one of the previously covered basic functions (such as squaring and reciprocal).

Examples of Basic Exponential Functions

2xf x 5xf x

0.5xf x 0.8xf x


Observations about exponential functions: 1. f(0) = 1 for all a > 0. (Basic exponential

functions have y-intercepts of 1). 2. When a > 1, a basic exponential function is

always increasing. When 0 < a < 1, a basic exponential function is always decreasing.

3. At one end of the function displays asymptotic

behavior dependent on the conditions in #2. 4. The closer a is to 1, the more linear the graph

appears.

End behavior of a basic exponential function

Recall that end behavior is the term used to describe what happens to function values as x gets very large positive (x → ¥) or very large negative (x → −¥). Further recall that an asymptote is a line that a graph approaches but never intersects.

Exponential functions and their end behavior can be divided into two groups based on the value of a: When a > 1, the exponential function will increase without bound as x → ∞ and will asymptotically approach 0 as x → −∞.

When 0 < a < 1, the exponential function will asymptotically approach 0 as x → ∞ and will increase without bound as x → −∞.


The number e

Consider the special expression 11 .n

n

We know that, for any exponential expression with base greater than 1, the value of the expression will grow rapidly as n gets very large positive.

We also know that 1 to any power is 1.

So what will happen to this expression as n gets very large? Will the exponent being very large dominate and send the value toward infinity or will the base being very close to 1 dominate and send the value toward 1?

Definition:

1lim 1 2.71828182846n

ne

n

e is called Napier’s Constant, Euler’s Number, or the Natural Base. Definition: The natural exponential function is

an exponential function with base e: .xf x e

n 1

1n

n


Exponential Functions based on limited information

A limited amount of information is necessary to graphically define an exponential function. You need to know if the function has been dilated, what its y-intercept is, and one point other than the y-intercept.

Ex. 1 Find the exponential function of the form xf x a defined by the graph.

Ex. 2 Find the exponential functions of the form xf x Ca defined by the graph.

Ex. 3 Find the exponential functions of the

form xf x Ca defined by the graph.

(0,1) 15( 1, )

(0,4)

( 2,1)

(0,3)( 1,5)


Transformations of basic exponential functions

The algebraic transformations we covered in a previous unit are possible with exponential functions as well.

Ex. 1 Identify the basic function in the given function.

Determine the transformations applied to the basic function that produce the given function. State the transformations applied, in the correct order, using units and directions as appropriate.

Finally, determine the domain, range, and asymptote of the given function.

2( ) 3 2xf x

Ex. 2 Identify the basic function in the given function.

Determine the transformations applied to the basic function that produce the given function. State the transformations applied, in the correct order, using units and directions as appropriate.


1( ) 6 3xf x


Topic 37: Compound Interest There are two primary formulas used for calculating the value of an account where interest is compounded:

Formula 1 – For Discretely Compounded Interest

1n trA t P

n

Formula 2 – Continuously Compounded Interest r tA t P e t = units of time in years P = principle (initial value) r = rate of interest as a decimal A(t) = amount at/after time t n = number of compounds per year

Ex. 1 Josh wishes to invest $2 500. He is looking at two different Certificates of Deposit (CDs), each of which compounds interest monthly. The two year CD earns 1.20% interest while the five year CD earns 2.25% interest. How much will each CD be worth at maturation? Express to the penny.


Ex. 2 Catie will be saving $10 000 for 10 years in three accounts. Three banks, A, B, and C, each offer the same 2.99% interest rate. Bank A compounds quarterly. Bank B compounds weekly. Bank C compounds continuously. How much will Catie earn in each account? Express to the peny.

Ex. 3 A furniture company offers a promotion where no interest is accrued on an account for four years. In the fine print you find that if the account is not paid in full within the four year period, all of the interest that could have accumulated on the original amount will retroactively be applied to the account. A $3 000 sofa is not paid off in time. How much interest would be tacked on to the account if the interest is 15.99% compounded biweekly? Express to the dollar.


Annual Percentage Rate Since accounts with the same interest rate can result in different amounts based on the frequency of compounding, a common tool for comparing rates is the Annual Percentage Rate (APR). To determine the APR on an account, calculate the value of investing $1 for 1 year and then subtracting the $1 before expressing as a percent. Ex. 4 A bank offers a nominal rate of 3.25%,

compounded daily. What is the APR on the loan? Express to the hundredth of a percent.

Present Value of a Future Amount Definition: The amount the principle must be to

achieve a desired value at time t. Ex. 5 What is the present value of of $5 000

for an account bearing 7.5% interest compounded monthly for 8 years? Express to the dollar.


Topic 38: Other Applications of Exponentials A variety of application problems can use formulas with exponentials in them. Formula 1: Annual Population Growth

The growth (or decline of a population) can be determined by the formula 0 1 .tn t n r

n0 = initial population n(t) = population at time t t = time in years r = annual rate of growth Formula 2: Growth based on a continuous rate

The growth (or decline of a population) can be determined by the formula 0 .r tn t n e

n0 = initial population n(t) = population at time t t = time (any units) r = relative rate of growth per unit of time

Formula 3: Radioactive Decay The amount of radioactive material in a sample can be determined by one of the equivalent formula

1

0 0 22 or .tthhm t m m t m

m0 = initial amount of radioactive material m(t) = amount of radioactive material at time t t = time h = half-life of radioactive material (units for time and half-life must agree) Formula 4: Newton’s Law of Cooling The temperature of an item after being in an environment of a different temperature can be found by the formula 0 .ktT t A T A e

t0 = initial item temp. m(t) = temperature at time t A = temperature of the environment t = time (any units) k = cooling rate


Ex. 1 An abandoned island, overrun with rats, is being used as a dumping ground. The rat population is given by the function 45 1.15 ,tn t where t is the number of years after 2005 and n(t) is in thousands.

a: According to the formula, what was the

population of rats on the island in 2005? b: According to the formula, what is the

growth rate in the rat population? c: Estimate the number of rats on the

island in the year 2020. Express to the tenth of a thousand.

Ex. 2 The population of Georgia on July 1, 2014 was approximately 10.1 million people and the population is expected to grow at an annual rate of 1.03% per year for the near future.

a: Write a function to describe the

population t years after 2014, letting population be measured in millions.

b: Estimate the population of fruit flies at 4

pm of the same day. Express to the nearest tenth of a million.


Ex. 3 A scientist wishes to do an experiment with fruit flies. At 9 am she has 200 flies and under the controls of her experiment their population should increase at a relative rate of 35% per hour.

a: Write a function to describe the

population t hours after 9 am. b: Estimate the population of fruit flies at 4

pm of the same day. Express to the whole fly.

Ex. 4 Modern smoke detectors use gamma rays produced by Americium-241 to help detect smoke at earlier stages. The half-life of Am-241 is 432 years. If 0.5 grams of Am-241 is used, how much will remain after 10 years when the life of the smoke detector is exhausted (due to mechanical and environmental wear)? Approximate to three decimal places.


Ex. 5 A new treatment for cancer that targets bone metastasis involves the use of Radium-223. The half-life of Ra-223 is 11.43 days. What percentage of Ra-223 remains in the body after 28 days? Express to the tenth of a percent.

Ex. 6 A cup of coffee brewed at 200ºF is in a room with a temperature of 70ºF. The cooling rate for the cup, with respect to time in minutes, is k = 0.04. Find the model for the temperature of the coffee, where time is in minutes, and then determine the temperature of the coffee an hour later.


Topic 39: Logarithms So what is the inverse of an exponential function?

It should be easy to determine the inverse of each of these functions: 3f x x 7g x x 3h x x Yet, we don’t have anything previously covered that serves as a proper inverse function to:

2xf x

Definition: Let a be a positive real number other than 1. Then the logarithm with base a, denoted as loga, is defined as follows:

log ya x y a x

Observations: A logarithm with base a is the

inverse operation of an exponential base a.

The output of a logarithmic expression is an exponent. Better said, a logarithmic expression is an exponent.


Evaluate these basic logarithmic expressions by considering their analogues amongst exponential expressions:

Ex. 1a 7log 1 Ex. 1b 11log 11 Ex. 1c 6log 36 Ex. 1d 3log 81

Properties of Logarithms 1. log 1 0a 2. log 1a a 3. log n

a a n 4. loga na n


Evaluate these basic logarithmic expressions by applying the properties of logarithms and properties of exponents:

Ex. 2a 2log 32 Ex. 2b 25log 5 Ex. 3a 9log 27 Ex. 3b 8log 4

Evaluate these basic logarithmic expressions by applying the properties of logarithms and properties of exponents:

Ex. 4a 112 12log

Ex. 4b 1

4 64log Ex. 5a 0.5log 16 Ex. 5b 5log 0.04


Special Logarithms Common Logarithms: Base 10 logarithm

10log log

log 10y

x x

y x x

Applications include pH scale, Richter scale, decibel scale, etc.

Natural Logarithms: Base e logarithm

log ln

lne

y

x x

y x e x

Since the natural exponential function has so many uses, its inverse should logically have many uses as well.

Properties of Special Logarithms Common Natural 1. log 1 0 ln 1 0 2. log 10 1 ln 1e 3. log 10n n ln ne n 4. log10 n n ln ne n


Using the properties of logarithms it should be relatively easy to evaluate 2log 32. Calculators will easy handle approximating log32 and ln32. But how do we determine the approximate value of

3log 32? At the very least we should be able to determine between which two integers the value should fall.

Change of Base Formula

logloglog

ba

b

mma

This formula allows us to calculate the approximate value of logarithms where the exact value is difficult or impossible to express as a non-logarithmic expression. Since we can choose what we change the base into, the most logical choices are usually base 10 or base e, thus:

ln loglogln loga

m mma a

Ex. 6 Approximate the value of 3log 32, five

places after the decimal point.


Topic 40: Logarithmic Functions & their Transformations

Basic logarithmic functions are logically the inverse functions to corresponding basic exponential functions. Compare the graphs of the exponential function base two with the logarithmic function base two.

Definition: For a logarithmic function log ,af x x the domain is (0, ∞),

the range is (−∞, ∞), and a vertical asymptote exists at x = 0.

Most programs for graphing will only allow for inputs using common or natural logarithms. Recall that a logarithmic function of the form logaf x x

can be expressed as lnln

xf xa

or loglog

xf xa

using the Change of Base Formula.

1 2xf x

2logf x x

y x


Transformations are also possible with logarithmic functions.

Ex. 1 Identify the basic function used.

Then determine the transformations applied to the basic function that produce the given function. State the transformations applied, in the correct order, using units and directions as appropriate.


4( ) log 2f x x

Ex. 2 Identify the basic function used.

Then determine the transformations applied to the basic function that produce the given function. State the transformations applied, in the correct order, using units and directions as appropriate.


3( ) 5log 1f x x


Logarithmic Functions based on limited information

Similar to exponential functions, it is possible to graphically define a logarithmic function using limited information. Assuming the logarithmic function has not be dilated, you need to know one point other than the x-intercept (1,0) to find the function.

Ex. 3 Find the logarithmic function of the form log af x xdefined by the graph.

Ex. 4 Find the logarithmic function of the form log af x xdefined by the graph.

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