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Chapter 1 Functions and Linear Models Sections 1.1 and 1.2

Chapter 1 Functions and Linear Models Sections 1.1 and 1.2. Functions and Linear Models. Functions: Numerical, Algebraic Functions: Graphical Linear Functions Linear Models. Functions.

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Chapter 1 Functions and Linear Models Sections 1.1 and 1.2

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  1. Chapter 1Functions andLinear ModelsSections 1.1 and 1.2

  2. Functions and Linear Models • Functions: Numerical, Algebraic • Functions: Graphical • Linear Functions • Linear Models

  3. Functions A real-valued function f is a rule that assigns to each real number x in a set X of numbers, a unique real number y in a second set Yof numbers. The set X is called the domain of the function f and the second set Y is called the codomain of f.

  4. Functions For each element x in the domain Xof the function, the corresponding element y in Y is called the image of x under the function f. The image is denoted by f (x), that is, y = f (x). f (x) is read “f of x.” The set of all images of the elements of the domain is called the range of the function.

  5. Not in the range of f A way to picture a function is by an arrow diagram f x y x y x X Y DOMAIN RANGE

  6. Functions • A symbol that represents an arbitrary number in the • domain of a function f is called an independent • variable. A symbol that represents a number in the • range of f is called a dependent variable. • A function can be specified: • algebraically:by means of a formula • numerically:by means of a table • graphically:by means of a graph

  7. Note on Domains The domain of a function is not always specified explicitly. If no domain is specified for the function f, we take the domain to be the largest set of numbers x for which f (x) makes sense. This "largest possible domain" is sometimes called the natural domain.

  8. Algebraically Defined Function Is a function represented by a formula. It has the format y = f (x) = “expression in x” is a function. Example: Substitute 5 for x Substitute x+h for x

  9. Algebraically Defined Function Is a function represented by a formula. It has the format y = f (x) = “expression in x” is a function. Example: In this case the natural domain of the function is the set of all real numbers. That is, Domf = (– , )

  10. Algebraically Defined Function is a function. Example: In this case the natural domain of the function is the set In interval notation this is

  11. Algebraically Defined Function is a function. Example: In this case the natural domain of the function consists of all values of z such that In interval notation this is

  12. Numerically Specified Function This is the case when we give numerical values for the function (the outputs, say the y-values) for certain values of the independent variable, say x. In this case the function is represented by a table which looks like.

  13. Numerically Specified Function Example: Suppose that the function f is specified by the following table. Then, f (0) is the value of the function when x = 0. From the table, we obtain f (0) = 3.01 Look on the table where x = 0 f (1) = -1.03 Look on the table where x = 1 and so on

  14. Numerically Specified Function • Example: The human population of the world P depends on the time t. • The table gives estimates of the world • population P(t) at time t, for certain • years. For instance, • However, for each value of the time t, • there is a corresponding value of P, • and we say that P is a function of t.

  15. Numerically Specified Function Example: The data represents the velocity V of an object, in feet/sec, after t seconds have elapsed. Note:at 2 seconds the object is going at 4.9ft/sec, that is V(2) = 4.9 ft/sec. The table can be represented graphically as follows

  16. Numerically Specified Function

  17. Mathematical Modeling To mathematically model a situation means to represent it in mathematical terms. The particular representation used is called a Mathematical model of the situation. Mathematical models do not always represent a situation perfectly or completely. Some represent a situation only approximately, whereas others represent only some aspects of the situation.

  18. Mathematical Modeling Example: The monthly payment, M, necessary to repay a home loan of P dollars, at a rate of r % per year (compounded monthly), for t years, can be found using

  19. Mathematical Modeling Example: A farmer has 1000 yards of fencing to enclose a rectangular garden. Express the area A of the rectangle as a function of the width x of the rectangle. What is the domain of A?

  20. Mathematical Modeling Example:Human population The table shows data for the population of the world in the 20th century. The figure shows the corresponding scatter plot.

  21. Mathematical Modeling The pattern of the data points suggests exponential growth.

  22. Mathematical Modeling We use a graphing calculator with exponential regression capability to apply the method of least squares and obtain the exponential model

  23. Mathematical Modeling We see that the exponential curve fits the data reasonably well. The period of relatively slow population growth is explained by the two world wars and the Great Depression of the 1930s.

  24. Use when x values satisfy condition 1 Use when x values satisfy condition n Piecewise Defined Functions Sometimes we need more than a single formula to specify a function algebraically. In this case the formula used to evaluate the function depends on the value of x.

  25. Use when x values are less than or equal to 2 Use when x values are greater than 2 Piecewise Defined Functions The following is a quick example of a piecewise defined function Notice = 26.5 = 53.8

  26. Piecewise Defined Functions The following is a quick example of a piecewise defined function Notice that the domain of f , in this case, is the set all real numbers. That is, Domf = (– , )

  27. Piecewise Defined Functions The percentage p(t) of buyers of new cars who used the Internet for research or purchase since 1997 is given by the following function.† (t = 0 represents 1997). Notice that the domain of p is the interval [0 , 4]. That is, Domp = [0 , 4]. †The model is based on data through 2000. Source: J.D. Power Associates/The New York Times, January 25, 2000, p. C1

  28. Piecewise Defined Functions This notation tells us that we use the first formula, 10t + 15, if 0  t < 1, or, t is in [0, 1) the second formula, 15t + 10, if 1  t  4, or, t is in [1,4]

  29. Piecewise Defined Functions Thus, for instance, p(0.5) = 10(0.5) + 15 = 20 Here we used the first formula since 0  0.5 < 1, or, equivalently, 0.5 is in [0, 1). p(2) = 15(2) + 10 = 40 We used the second formula since 1  2  4, or equivalently, 2 is in [1, 4]. p(4.1) is undefined p(t) is only defined if 0  t  4.

  30. Graphically Specified Function The most common method for visualizing a function is its graph. The graph of a function is the set of all points (x, f (x)) in the xy-plane such that x is in the domain of f . Sometimes the function is only known through its graph and may be very difficult to represent it algebraically. The next example illustrates this case.

  31. Graphically Specified Function The vertical acceleration a of the ground as measured by a seismograph during an earthquake is a function of the elapsed time t. The figure shows a graph generated by seismic activity during the Northridge earthquake that shook Los Angeles in 1994. For a given value of t, the graph provides a corresponding value of a.

  32. Graphically Specified Function Example: The monthly revenue R from users logging on to your gaming site depends on the monthly access fee p you charge according to the formula (R and p are in dollars.) Sketch the graph of R. Find the access fee that will result in the largest monthly revenue.

  33. Graphically Specified Function Solution: To sketch the graph of R by hand, we plot points of the form (p,R(p)) for several values of p in the domain [0,2.5] of R. First, we calculate several points.

  34. Graphically Specified Function Graphing these points gives the graph in the figure on the left, suggesting the parabola shown on the right.

  35. Graphically Specified Function The revenue graph appears to reach its highest point when p = 1.25, so setting the access fee at $1.25 appears to result in the largest monthly revenue.

  36. Graphically Specified Function Example: The following table gives the weights (in pounds) of a particular child at various ages (in months) in her first year.

  37. If we represent the data given in the table graphically by plotting the given pairs (t ,W(t)), we get, (we have connected successive points by line segments) W(5) = 16 W(4.5) 

  38. Graphically Specified Function Example: Given the graph of y = f (x), find f (1). f (1) = 2 (1, 2)

  39. Graphically Specified Function Using the definition of graph of a function and the calculations done in the previous examples we can now see how to determine the domain and range of a graphically defined function.

  40. y 4 (2, 3) (10,0) 0 x (4, 0) (1, 0) (0, -3) -4 Example: Determine the domain, range, and intercepts of the function defined by the following graph.

  41. Graph of a Function Vertical Line Test: The graph of a function can be crossed at most once by any vertical line. Function Not a Function It is crossed more than once.

  42. y x

  43. y x

  44. Sketching a Piecewise Function Sketch the portion of the formula on its domain

  45. Useful Functions and Their Graphs

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