DEFINITION Function A function from a set D to a set R is a rule that assigns a unique element in R to each element in D.
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1 AP Calculus Assignment #2; Functions and Graphs Name: Functions The values of one variable often depend on the values for another:! The temperature at which water boils depends on elevation (the boiling point drops as you go up).! The amount by which your savings will grow in a year depends on the interest rate offered by the bank.! The area of a circle depends on the circle s radius. In each of these examples, the value of one variable quantity depends on the value of another. For example, the boiling temperature of water, b, depends on the elevation, e; the amount of interest, I, depends on the interest rate, r. We call b and I dependent variables because they are determined by the values of the variables e and r on which they depend. The variables e and r are independent variables. A rule that assigns to each element in one set a unique element in another set is called a function. The sets may be sets of any kind and do not have to be the same. A function is like a machine that assigns a unique output to every allowable input. The inputs make up the domain of the function; the outputs make up the range. DEFINITION Function A function from a set D to a set R is a rule that assigns a unique element in R to each element in D. Euler invented a symbolic way to say y is a function of x ;, which we red as y equals f of x. This notation enables us to give different functions different names by changing the letters we use. To say that the boiling point of water is a function of elevation, we can write. To say that the area of a circle is a function of the circle s radius we can write, giving the function the same name as the dependent variable. 1. Write a formula that expresses the area of a circle as a function of its radius. Use the formula to find the area of a circle of radius 2 in. 2. Write a formula for the surface are S of a cube as a function of the length of the cube s edge e; then use your formula to find the surface area of a cube of edge length 5 ft. Domains and Ranges In the you try above, the domain of the function is restricted by context; the independent variable is the radius and must be positive. When we define a function with a formula and the domain is not stated explicitly or restricted by context, the domain is assumed to be the largest set of x-values for which the formula gives real y-values the so-called natural domain. If we want to restrict the domain, we must say so. The domains and ranges of many real-valued functions of a real variable are intervals or combinations of intervals. The intervals may be open, closed, or half-open and finite or infinite.
2 The endpoints of an interval make up the interval s boundary and are called boundary points. The remaining points make up the interval s interval and are called interior points. Closed intervals contain their boundary points. Open intervals contain no boundary points. Every point of an open interval is an interior point of the interval. Viewing and Interpreting Graphs The points (x,y) in the plane whose coordinates are the input-output pairs of a function the function s graph. The graph of the function (x,y) for which y equals x + 2. Example Identifying Domain and Ranges of a Function Identify the domain and range and then sketch a graph of the function. make up, for example, is the set of points with coordinates Solution The formula gives a real y-value for every real x-value except x = 0. The domain is. The value y takes on every real number except y = 0. The range is also below.. A sketch is shown The formula gives a real number only when x is positive or zero. The domain is denotes the principal square root of x, y is greater than or equal to zero. The range is also is shown below.. Because. A sketch Graph Viewing Skills 1. Recognize that the graph is reasonable. 2. See all the important characteristics of the graph. 3. Interpret those characteristics. 4. Recognize grapher failure. Being able to recognize that a graph is reasonable comes with experience. You need to know the basic functions, their graphs, and how changes in their equations affect the graphs. Grapher failure occurs when the graph produced by a grapher is less than precise or even incorrect usually due to the limitations of the screen resolution of the grapher, or the window provided by the user. Use a grapher to identify the domain and range of the following functions.
3 Even Functions and Odd Functions Symmetry The graphs of even and odd functions have important symmetry properties. DEFINITIONS Even Function, Odd Function A function is an for every x in the function s domain. even function of x if odd function of x if The names even and odd come from powers of x. If y is an even power of x, as in, it is an even function of x (because ). If y is an odd power of x as in, it is an odd function of x (because ). The graph of an even function is symmetric about the y-axis. Since, a point (x,y) lies on the graph if and only if the point (-x,y) likes on the graph. The graph of an odd function is symmetric about the origin. Since, a point (x,y) lies on the graph if and only if the point (-x,-y) lies on the graph. Equivalently, a graph is symmetric about the origin if a rotation of 180 about the origin leaves the graph unchanged. Determine whether the function is even, odd, or neither. Try to answer without writing anything, except the answer. Justify your answer. (c) (d) (e) Functions Defined in Pieces While some functions are defined by single formulas, others are defined by applying different formulas to different parts of their domains. 1. Graph the functions below.
4 2. Let f be the function defined on the closed interval [-5,4]. The graph of f consists of three line segments and one semicircle as shown above. a. Write the piecewise function for. b. Find the values of x for which. c. For what intervals of x does have a negative rate of change? Explain your answer. d. On what intervals of x is the rate of change of the greatest? What is that rate and justify your answer. Absolute Value Function The absolute value function is defined piecewise by the formula. The function is even, and its graph is symmetric about the y-axis. The absolute value function has domain and range. From this given graph sketch the graph of the domain and range of the transformed graph.. Then find Solution: The graph of f is the graph of the absolute value function shifted 2 units horizontally to the right and 1 unit vertically upward. The domain is and the range is.
5 Draw the graph of the function then find its domain and range. Composite Functions Suppose that some of the outputs of a function g can be used as inputs of a function f. We can then link g and f to form a new function whose inputs x are inputs of g and whose outputs are the numbers. We say that the function (read f of g of x) is the composite of g and f. It is made by composing g and f in the order of first g, then f. The usual stand-alone notation for this composite is read as f of g. Thus, the value of at x is., which is Find a formula for if and. Then find. Exploration: Composing Functions Some graphers allow a function such as such a grapher, we can compose functions. to be used as the independent variable of another function. With 1. Enter the functions and. Which of and corresponds to? to? 2. Graph and and make conjectures about the domain and range of. 3. Graph and and make conjectures about the domain and range of. 4. Confirm your conjectures algebraically by finding formulas for and.
6 Practice; Use your own paper. 1. The table below shows the gross revenue for the Broadway season in millions of dollars for several years. Year Amount ($millions) Find the quadratic regression for the data in the table. Let x = 0 represent Superimpose the graph of the quadratic regression equation on a scatter plot of the data (c) Use the quadratic regression to predict the amount of revenue in (d) Now find the linear regression for the data and use it to predict the amount of revenue in (e) Which is a better predictor of revenue for 2008, why? 2. Dayton Power and Light, Inc. has a power plant on the Miami river where the river is 800 ft wide. To lay a new cable from the plant to a location in the city 2 mi downstream on the opposite side costs $180 per foot across the river and $100 per foot along the land.(picture is not drawn to scale, or drawn very well.) 800 ft P x Q Dayton Power Plant Suppose that the cable goes from the plant to a point Q on the opposite side that is x ft from the point P directly opposite the plant. Write a function that gives the cost of laying the cable in terms of the distance x. Generate a table of values to determine if the least expensive location for point Q is less than 2000ft or greater than 2000 ft from point P. (c) Can you find where they should locate point Q in order to minimize the cost of the cable? If so, where? 3. Which of the following gives the domain of? (c) (d) (e) 4. If and, which of the following gives? 2 6 (c) 7 (d) 9 (e) Must the product of two even functions always be even? Give reasons for your answer. 6. Can anything be said about the product of two odd functions? Give reasons for your answer. 7. The pollutant PCB (polychlorinated biphenyl) affects the thickness of pelican eggs. Thinking of the thickness, T, of the eggs, in mm, as a function of the concentration, P, of PCBs in ppm (parts per million),
7 we have. Explain the meaning of in terms of thickness of pelican egs and concentration of PCBs. 8. The value of a car,, in thousands of dollars, is a function of the age of the car, a, in years. Interpret the statement using a complete sentence. Sketch a possible graph of V against a. Is f an increasing or decreasing function? Explain. (c) Explain the significance of the horizontal and vertical intercepts in terms of the value of the car. 9. An object is put outside on a cold day at time t = 0. Its temperature, in is decreasing at a decreasing rate (i.e. it is concave up). What does the statement mean in terms of temperature? Include units for 30 and for 10 in your answer. Explain what the vertical intercept and the horizontal intercept represent in terms of temperature of the object and time outside. 10. You drive at a constant speed from Chicago to Detroit, a distance of 275 miles. About 120 miles from Chicago you pass through Kalamazoo, Michigan (yes this is a real city). Sketch a graph of your distance from Kalamazoo as a function of time. 11. The monthly charge for a waste collection service is $32 for 100 kg of waste and is $48 for 180 kg of waste. Find a linear formula for the cost, C, of waste collection as a function of the number of kilograms of waste, w. What is the slope of the line found in part? Give units with your answer and interpret it in terms of the cost of waste collection. (c) What is the vertical intercept of the line found in part? Give units with your answer and interpret it in terms of the cost of waste collection. 12. If, find the domain of. Solve. 13. If, find all values of t for which is a real number. Solve. 14. The demand function for a certain product,, is linear, where p is the price per item in dollars and q is the quantity demanded. If p increases by $5, market research shows that q drops by two items. In addition, 100 items are purchased if the price is $550. Find a formula for q as a linear function of p. Find a formula for p as a linear function of q. (c) Draw a graph with q on the horizontal axis. 15. A body of mass m is falling downward with velocity v. Newton s Second Law of Motion, says that the net downward forces, F, on the body is proportional to its downward acceleration, a. The net for, F, consists of the force due to gravity,, which acts downward, minus the air resistance,, which acts upward. The force due to gravity is mg, where g is a constant. Assume the air resistance is proportional to the velocity of the body. Write an expression for the net force, F, as a function of the velocity v. Write a formula giving a as a function of v. (c) Sketch a against v.
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