Element x in D is called the input or the independent variable of the function.
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1 P a g e 1 Chapter 1. Functions and Mathematical Models Definition: Function A function f defined on a collection D of numbers is a rule that assigns to each number x in D a specific number f(x) or y. We say that y is a function of x. Element x in D is called the input or the independent variable of the function. Element f(x) or y is called the output or the dependent variable of the function. Definition: Domain The collection (or set) D of all numbers for which f(x) is defined is called the domain of the function f. It consists all possible inputs. Definition: Range The set of all possible values y=f(x) is called the range of the function. It consists all possible outputs. Functions may be given in the form of Diagrams, Charts, Tables, Graphs, Formulas or Words. Three questions related to the function:
2 P a g e 2 1. Is it a function? To answer this question, you need to know which one is the dependent variable and which one is the independent variable. a. Is y a function of x? b. Is x a function of y? Questions a and b are two different questions. 2. Finding Input and Output Values 3. Finding Domain and Range Section 1.1 Functions defined by tables Example 1 on page 4: The following table gives the average price of a 30-second commercial airing during the Super Bowl in the indicated year. Year Price of Commercial(in millions) 1998 $ $ $ $ $2.4 a. Is It a Function? i) Is the price of commercial a function of the year? ii) Is the year a function of the price of the commercial? Example 2: Now we know that the price of commercial is a function of the year. b. Find Input and Output Values. i) What is the price of commercial in 2001? ii) In which year(s) the price of commercial is 2.1 million dollars? c. Domain and Range i) What is the domain of the function? ii) What is the range of the function?
3 P a g e 3 Section 1.2 Functions defined by graphs In the graph, we always put the independent variable or input on the horizontal axis and put the dependent variable or output on the vertical axis. So we have the vertical line test. In order for a graph to represent a function, any vertical line must cross the graph at most once. Example 1 on page 10: The following scatter plot illustrates the population (in thousands) of St. Louis, Missouri, for the census years a. Does this scatter plot represent a function? b. Find population for St. Louis in year c. Find population for St. Louis in year d. Find domain of the function. e. Find range of the function.
4 P a g e 4 Example 4 on page 13: The following figure shows the typical distance (in feet) that a car travels after the brakes have been applied for various speeds (in miles per hour). a. Does this graph represent a function? b. Find the distance a car traveled after the brakes have been applied if its speed is 60 miles per hour. c. Find the distance a car traveled after the brakes have been applied if its speed is 55 miles per hour. d. Find domain of the function. e. Find range of the function. Section 1.3 Functions defined by formulas Example 1 on page 19. In June 2005, T-mobile advertised a Get More Plan for cell phone service that included 600 whenever minutes plus unlimited weeknight and weekend minutes each month for $39.99, with additional minutes charged at 40 cents each. Let s assume that only weekday minutes count as whenever minutes against the 600-minute total under the $39.99 basic monthly charge. a. Does this formula describe the monthly cost of cell phone service as a function of the number of weekday minutes used?
5 P a g e 5 b. Write a symbolic function C(n) = giving the monthly cost of cell phone service as a function of the number of weekday minutes used n. Example 2. a. Is y a function of x? b. Is x a function of y? Example 3. a. Find b. Find a such that Example 4. Find domain for the following functions: a. b. c.
6 P a g e 6 d. e. Section 1.4. Average Rate of Change A function is increasing over the interval of x-values if, for any two different values and in the interval, if < then. A function is decreasing over the interval of x-values if, for any two different values and in the interval, if < then. A function is constant over the interval of x-values if, for any two different values and in the interval,. Definition: Average rate of change of a function y=f(x) over the interval [a,b] is defined by f ( b) f ( a) b a
7 P a g e 7 Example 2 on page 28. The following table gives the percentage of the U.S. population that was born outside of the United States as a function of the year. a. Use the table to determine the intervals where the function is increasing, decreasing, or constant. b. Find the average rate of change of f(x) over each interval of consecutive x-values. Year x Percentage of Population Born Outside U.S. f(x) Average Rate of Change You may also calculate the average rate of change by using calculator: Average Rate of Change Store x-value in L 1 and y-value in L 2. Highlight L 3 and enter Δlist(L 2 )/ Δlist(L 1 ) (to get Δlist: 2 nd Stat, OPS, 7)
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