1. Function Notation
Evaluating a function means figuring out the value of a function’s output from a particular value of the input.
Example. Let the function g be defined by: Evaluate g(3) = ((3)2+1)/(5+3) = 10/8 = 1.25

2. Evaluating functions using a table
Suppose that f is defined by the table:
To find f(3), we look in the table and get f(3) = –4.
Now define g(x) = f(x+1). We evaluate g(3) = f(4) = –3. The table for g appears below:
Why is no value listed for g(4)? Why is g defined at x= –2 while f is not?
x –
f(x) – – – 3
x – –
g(x) – – –

3. Given an input, we evaluate a function to find the output
Given an input, we evaluate a function to find the output. Often the situation is reversed; we know the output value and we want to find the corresponding input value(s). If the function is given by a formula, the input values are solutions to an equation.
Problem. Let A = f(r) be the area of a circle of radius r, where r is in cm. What is the radius of a circle whose area is 100 cm2 ? Solution. The output f(r) is an area. Solving the equation f(r) = 100 for r gives us the radius of a circle whose area is 100 cm2. Since the formula for the area of a circle is we solve This yields and we take the positive value.

4. Finding the input given the output value using a table
Suppose that f is defined by the table:
As before, to find f(1), we look in the table and get f(1) = –5.
Now suppose we want to solve f(x) = 2 for x.
There are two values for x which satisfy this condition,
namely, x = –1 and x = 2.
x –
f(x) – – – 3

5. Finding the input given the output value using a graph
y = f(x)
If f(x) = 2, then x = 1 or x = 3.

6. Changes in input and output
For the function Q = f(t), a change inside the function’s parentheses can be called an “inside change” and a change outside the function’s parentheses can be called an “outside change”.
Problem. For the function, which represents the area of a circle of radius r, contrast the expressions f(r + 10) and f(r) Solution f(r + 10) is the area of a circle of radius = (r + 10) f(r) + 10 is the area of a circle of radius = r plus 10 more units of area.

7. Domain and Range
Assume that Q = f(t) The domain of f is the set of input values, t, which yield an output value The range of f is the corresponding set of output values.
There are two types of reasons that the domain of a function f may be restricted: (1) The real-world situation being modeled by f does not make sense without a restriction (2) The formula used to define f does not make sense without a restriction (often because of division by 0).
If the domain of a function is not specified, we usually assume that it is as large as possible--that is, all numbers that make sense as inputs for the function.

8. Problem. Find the domain and range of. Solution
Problem. Find the domain and range of Solution. Since no restrictions on the domain are given, we make it as large as possible. That is, all real numbers except To determine the range, we suppose that y is in the range so that The latter equation can be solved for x if Also, if y = 2, the latter equation has no solution for x. Thus, the range of f is the set of all real numbers except 2.

9. Use of Maple to plot function of previous slide
> plot(2+(1/x),x=-3..3, ,color=black);

10. Finding the domain and range for a function whose graph is given

11. Model for height of a sunflower plant
The height of a certain sunflower in centimeters as a function of time in days is:
Assuming that the domain of h is , the graph of h is:
By examining the graph, can you tell what the range of h is?

12. Piecewise Defined Functions
A function may employ different formulas on different parts of its domain. Such a function is said to be piecewise defined. Note that we say there is one function even though several formulas may be used.
Example. A long-distance calling plan charges 99 cents for any call up to 20 minutes in length and 7 cents a minute for each additional minute with part of a minute pro-rated. Let C be the cost of a call in cents as a function of its length t in minutes. We have C = f(t), where the formulas for this function are: The graph of this function is shown on the next slide.

13. Cost Function for Long Distance Calling Plan
• Where is the point on the graph when t = 0?

14. The next problem introduces a piecewise defined function which is used to round a value x down to the nearest integer. How would you define the function that rounds a value x up to the nearest integer, and what would you call it?
Problem. For any real number x, we define the floor of x, denoted as the greatest integer that does not exceed x. Find the formulas for the floor of x when Solution Note that the floor function is first described in words, and this yields an infinite number of different formulas for the function.

15. Use of Maple to graph the floor function
>plot(floor(x),x= ,color=black,discont=true);
What does your calculator call the floor function?

16. The Absolute Value Function
The absolute value function is a piecewise defined function defined by
The graph of y = | x | is shown below:
The absolute value function is used to measure the distance |x–y| between points x and y on the number line. For example, the distance from –1 to 2 is | –1 –2| = | –3| = 3.

17. What are the values of p1 and p2?
Suppose an employee is paid $5.00 per hour to work a standard 40 hour work week. If he works overtime, he is paid $7.50 per hour up to a maximum of 80 hours. The graph below shows his weekly pay as a function, f(t), of the time worked. p2 p1
What are the values of p1 and p2?

18. Pay function in bracket form
(80,500)
(40,200)
f(t) =

19. Composite Functions
Two functions may be combined by requiring the output of one to be the input of the other.
For example, let n = f(A) =A/250 be the number of gallons of paint needed to paint A square feet, and let C = g(n) = 30.5n be the cost, in dollars, of n gallons of paint.
Substituting n into the formula for C, we obtain where the function h is said to be the composition of f and g.
We say that f is the inside function and g is the outside function.

20. Example of composite functions
Let f(x) = |x| – 1. The graph of f is shown next.
Let g(x) = |x|. Then h(x) = g(f(x)) = | |x| – 1 |. Then the graph of h(x) is:
Challenge: Draw the graph of h(h(x)).

21. Interchanging input and output: Inverse Functions
The roles of input and output are not necessarily fixed. In an earlier example, we derived a function f which converts degrees Celsius, C, to degrees Fahrenheit, F. The formula for f was: Suppose now that we know the value of F and we wish to compute the value of C. We can define a new function g such that C = g(F). For this function, F is the input and C is the output. The functions f and g are called inverses of each other.
Find a formula for the inverse function C = g(F). We solve the previous equation for C to obtain:

22. Inverse Function Notation
In the preceding discussion on temperature conversion, their was nothing about the names of the two functions that stressed their special relationship. If we want to emphasize that a function is the inverse of f, we call it f -1, which is read “f-inverse”.
For temperature conversion, we have:
Warning:

23. A function and its inverse "undo" each other.
What happens if we convert from Celsius to Fahrenheit and then back to Celsius? Answer: We are back where we started. In terms of function composition,
If, on the other hand, we convert from Fahrenheit to Celsius and then back to Fahrenheit, we have

24. Finding a formula for an inverse function
In the graph below, we have P = f(t) = t, and we want to graphically find
Locate 25 on the P-axis P
Read off the value of t corresponding to P = 25.

25. If we continue with the example from the previous slide, we may also solve algebraically for The equation to be solved is: t = 25, and we first subtract 20 from both sides to obtain 0.4t = 5. Upon dividing by 0.4, we have t =
This same algebraic procedure can be carried for a general P to obtain the formula for We must solve t = P for t. We have 0.4t = P – 20, and thus, That is,

26. Facts about Inverse Functions.
Suppose f has an inverse function. Then outputs of f are inputs of f -1. Similarly, outputs from f -1 are inputs of f. It follows that: Domain of f -1 = Range of f and Range of f -1 = Domain of f.
Warning: Not all functions have inverses. The functions which have inverses are called invertible.
Example. The absolute value function y = | x | is not invertible. Do you see why?

27. Concavity
Suppose we graph Q = f(t). If the graph bends upward as t increases, we say that f or its graph is concave up. If the graph bends downward as t increases, we say that f or its graph is concave down.
The average rate of change of f over small intervals is what determines the concavity of f • If the average rate of change of f increases, then f is concave up • If the average rate of change of f decreases, then f is concave down.
Recall Karim’s bike trip. What can you say about concavity for Karim? Was his average speed increasing or decreasing? What about Amanda’s bike trip?

28. Quadratic Functions
A quadratic function is a function with a formula given by the general form f(x) = ax2+bx+c, where a, b, c, are constants and
Some quadratic functions can be expressed in factored form f(x) = a(x – r)(x – s), where a, r, and s are constants and
The graph of a quadratic function is called a parabola.
Conversion from general form to factored form for a quadratic function is discussed on the next slide.

29. Finding the zeros of a Quadratic Function
Input values x which satisfy f(x) = 0 are called zeros of f.
Sometimes we can find the zeros of a quadratic by factoring. For example, if f(x) = x2 – x – 6, we have which becomes (x – 3)(x + 2) = 0, so the zeros are x = 3 and x = –2. (See Tools for Chap. 2, page 99, for more on factoring.)
If we look for the zeros of f(x) = ax2 + bx + c, then by the quadratic formula we have:
If b2 – 4ac < 0, then x1 and x2 are not real numbers and the graph does not cross the x-axis.

30. Concavity and Quadratic Functions
Since the graph of a quadratic f(x) is a parabola, we know from previous experience that it is either concave up for all x or it is concave down for all x.
Another way to interpret the previous statement is that the average rate of change for a quadratic f(x) is either increasing for all x or decreasing for all x. See the figure below.
slope negative
slope positive
slope zero

31. Summary for Functions, Quadratics, and Concavity
Evaluating a function means figuring out the value of a function’s output from a particular value of the input.
Sometimes we are given the output value and we must solve for the input value.
Important terms: “inside change” and “outside change”.
Assume that Q = f(t) • The domain of f is the set of inputs which yield an output value. • The range of f is the corresponding set of output values.
A function which uses different formulas in different parts of its domain is said to be piecewise defined.
If h(x) = g(f(x)), we say that h is the composition of f and g.
If we are given y = f(x), and the roles of inputs and outputs are reversed, we have the inverse function x = f -1(y). Warning: Not all functions have inverses.

32. Summary for Functions, Quadratics, and Concavity, continued
When a function is composed with its inverse, we have
Quadratic functions have at most 2 zeros which can be found by factoring or by the quadratic formula. Their graphs are either concave up or concave down for all x.

33. Exponential Functions
Exponential functions are functions that change at a constant percent rate. This percent rate will be called the growth rate when it results in an increase and the decay rate when it results in a decrease.
Example. If you have a starting annual salary of $40000, and you get a 6% raise each year, what will your resulting salary be in subsequent years? Here, the growth rate is 6% After 1 year, your salary is (1.06)(40000) = After 2 years, your salary is (1.06)(42400) = and so on After t years, your salary is (1.06)t(40000).

34. > plot(40000*(1.06)^t,t=0..20,color=black);
Use of Maple to graph the salary from the previous slide
> plot(40000*(1.06)^t,t=0..20,color=black);

35. Example. Suppose that you invest $10000 in the latest “dotcom” venture which allows two people anywhere to get married on the web. Unfortunately, you find that the value of your investment at the end of each year is 5% less than it was at the beginning of the year. Here, the decay rate is 5% After 1 year, the value is (0.95)(10000) = After 2 years, the value is (0.95)(9500) = and so on After t years, the value is (0.95)t(10000).

36. >plot(10000*(0.95)^t,t=0..80,color=black);
Use of Maple to graph the investment value of previous slide
>plot(10000*(0.95)^t,t=0..80,color=black);

37. For an exponential function, we have
For an exponential function, we have New amount = (growth factor) (Old amount) where the old amount is present at the beginning of a period and the new amount is present at the end of a period.
In the salary example, the growth factor is 1.06 per year.
In the investment example, the growth factor is 0.95 per year. Note that we use the term “growth factor” even though the value of the investment is decreasing or decaying.

38. The formula for an exponential function Q = f(t) is given by
The formula for an exponential function Q = f(t) is given by where the parameter a is the initial value of Q (at t = 0) and the parameter b is the growth factor. If a > 0, then b > 1 gives exponential growth and 0 < b < 1 gives exponential decay. The growth factor or base is given by where r is the decimal representation of the percent rate of change.
In the salary example, a = 40000, b = 1.06, and r = 0.06.
In the dotcom example, a = 10000, b = 0.95, and r = –0.05.

39. Comparing exponential and linear functions
For a table that gives y as a function of x with x constant: If the difference of consecutive y-values is constant, the table could represent a linear function If the ratio of consecutive y-values is constant, the table could represent an exponential function.
Example. Which function is linear? exponential? x
f(x)
g(x)

40. Finding a formula for an exponential function
To find a formula for the exponential function on the previous slide, we must determine the values of a and b in the formula g(x) = abx. From the table, we have
From the formula, we have
Thus,
Finally, we solve a( )20 = 1000 for a, which yields

41. Comparison of exponential and linear functions
The 19th century clergyman and economist, Thomas Malthus, assumed that the food supply increases linearly and the population increases exponentially. This assumption combined with the following fact led to a gloomy prediction of the future.
It can be shown that an exponentially increasing quantity will, in the long run, always outpace a linearly increasing quantity. For the example shown in the table below, the exponential outpaces the linear function at some t value between 4 and 5. t 4t 2t

42. Parameters for exponential and linear functions
The general formulas for exponential and linear functions both have two parameters. Hence, two data points are sufficient to determine a particular function of either type.
The two parameters for linear functions are the slope and the vertical intercept.
The parameters a and b in the formula abt determine the shape of the corresponding graph. In the current lecture, we assume that a > 0. A graph with a < 0 can be easily obtained by reflection across the t-axis of one of the graphs with a > 0. If a = 0, the formula yields the zero function. As we did before, we assume that b > 0. Note that the value of an exponential function at 0 is ab0 = a.

43. Graphs of exponential functions Q = f(t) = abt with b > 1
These functions are both increasing since their graphs rise as t increases. Note that the greater the value of b, the more rapidly the graph rises. These graphs satisfy and we say that the graph has the t-axis as a horizontal asymptote. Alternatively, we write
Q = 50(1.4)t
Q = 50(1.2)t

44. Graphs of exponential functions Q = f(t) = abt with 0 < b < 1
These functions are both decreasing since their graphs fall as t increases. Note that the smaller the value of b, the more rapidly the graph falls. These graphs satisfy and we again say that the graph has the t-axis as a horizontal asymptote. Alternatively, we write
Q= 50(0.8)t
Q= 50(0.6)t

45. Concavity and rates of change
Again consider the salary example.
Since the rate of change increases with time, the graph bends upward. We say such graphs are concave up.
larger increase
4 years
smaller increase
4 years

46. Concavity and rates of change, continued
Exponential functions Q = abt with a > 0 are all concave up. We can consider the graph showing the distance traveled by a certain bicycle rider as a function of time.
Since the bicycle rider’s rate of change decreases with time, the graph bends downward. We say such graphs are concave down.
distance (miles) 60
Smaller increase 50
1 hr 40 30
Larger increase 20
1 hr 10 1 2 3 4 5
time (hours)

47. Motivating the number e
Suppose we invest $1.00 at 100% interest once a year. At the end of the year, we have $2.00.
Suppose we invest $1.00 at 50% interest twice a year. At the end of the year, we have $2.25.
Suppose we invest $1.00 at 25% interest four times a year. At the end of the year, we have $
As the frequency of computation increases, the balance at the end of the year approaches $ , and this limiting value is referred to as the number e.

48. The number e = is an irrational number which is often used as the base of an exponential function. Base e is often referred to as the “natural base”.
Example. Solve graphically for x: ex = 100. ex

49. Continuous growth and the number e
If Q = abt, b > 0, this relation can be rewritten as: where ek = b. The value k is called the continuous growth rate.
The value of the continuous growth rate may be given as a decimal or a percent. If t is in years, for example, then the units of k are per year; if t is in minutes, then k is per minute.
Example. If you have a starting annual salary of $40000, and you get a 6% raise each year, what is the continuous growth rate of your salary, Q? We have Q = 40000(1.06)t, so a = and b = Using Maple, we can solve ek = 1.06 to get:

50. Compound Interest
What if your salary increase was not 6% compounded yearly, but 6% compounded over a shorter period? Then the formulas used to compute your salary might be:

51. Effective annual yield
When the interest (or salary increase) of an account is compounded more frequently than once a year, the account effectively earns more than the nominal rate, which is 6% in the salary example. The effective annual rate gives the interest (or salary increase) actually earned by the account.
Example. When your salary is compounded monthly (see previous slide), after one year you have: and the effective annual rate is 6.168%. When your salary is compounded continuously, after one year you have: and the effective annual rate is 6.184%.

52. General formulas for compounding
If interest at an annual (nominal) rate of r is compounded n times a year, then r/n times the current balance is added n times a year. Therefore, with an initial deposit of $P, the balance t years later is:
If interest on an initial deposit of $P is compounded continuously at an annual (nominal) rate of r, the balance t years later can be calculated using the formula:

53. Summary for exponential functions
Exponential functions are functions that change at a constant percent rate. The general formula for an exponential function is abt, where b > 0. The number b is known as the growth factor or as the base. Also, b = 1+r, where r is the decimal representation of the percent rate of change.
For a > 0, an exponential function is increasing if b > 1, and it is decreasing if 0 < b < 1.
Two data points determine a particular exponential function.
An exponentially increasing quantity will eventually outpace a linearly increasing quantity.
An exponential function Q = f(t) has the t-axis as a horizontal asymptote.
Graphs of exponential functions with a > 0 are concave up, while other (non-exponential) graphs may be concave down.
The irrational number e is the so-called natural base.
Interest may be compounded n times a year or continuously.