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Domain Rule Range Lesson 2.3 Suppose we had a box that would take an input, apply a rule, and spit out the output. This would be called a function box. This function box takes the name of a movie and spits out the first letter of the name. This function box takes a number and adds 1 to it. This function box takes a number and calculates its square root. The numbers that go into the function box are called the domain or the x values. The numbers that come out of the function box after the rule is applied are called the Domain range or y values. Some examples of rules could be Y = 2x - 2, or y = x2. Rule Range When the number going into the function box produces a unique value coming out of the box, we say that the relationship between the two numbers constitutes a “function”. This means that you can’t put 4 into the box and get a 6 out one time and a 10 out the next time. The rule is often written with “function notation”. This means that instead of writing y = 2x - 2, the rule is written f(x) = 2x - 2. Definition of a Function A function is a relationship between two variables such that each value of the first variable is paired with exactly one value of the second variable. How to tell if a relation between two numbers is a function. Ordered Pairs - A set of order pairs is a function if every x-value produces a unique y-value. {(1,5), (2,9), (3,12)} This is a function {(1,5), (2,9), (3,12), (1,7)} This is not a function because the xvalue 1 produces both 5 and 7 y-values How to tell if a relation between two numbers is a function. Table: For each x-value, there is exactly one value of y. Do the numbers in the table represent a function? Domain, x 1 2 3 4 5 6 52 Range, y -3.6 -3.6 4.2 4.2 10.7 12.1 52 This is a function because each xvalue has only one y-value. How to tell if a relation between two numbers is a function. Table: For each x-value, there is exactly one value of y. Do the numbers in the table represent a function? Domain, x 3 3 3 4 10 11 52 Range, y 7 8 10 42 34 18 52 This is not a function because the x-value 3 has three different y-values. How to tell if a relation between two numbers is a function. Mapping Diagram- a mapping diagram shows a function if each member of the domain goes to only one member of the range. Range Domain This is a -2 10 function 4 3 7 -7 Domain Range -2 4 7 10 3 -7 13 This is a not function because the 7 in the domain produces both a -7 and a 13 in the range How to tell if a relation between two numbers is a function. Graph - a graph shows a function if it passes the vertical line test. This means that a vertical line drawn anywhere on the graph will pass through only one point on the graph. Is this graph a function? This parabola IS a function. We could not draw a vertical line that intersected the graph in more than one place. Is this graph a function? This ellipse is NOT a function! All it took was one vertical line that intersected in more than one place for us to be sure. Which of these are functions? The first two are functions. In fact, the first graph is a linear function. There is only one line that is NOT a function. That's right! A vertical line is not a function. Weight For the data shown, is weight a function of height? NO Height Definition of Relation A relationship between two variables such that each value of the first variable is paired with one or more values of the second variable is called a relation. The domain is the set of all possible values of the first variable. The range is the set of all possible values of the second variable. State the domain and range: Domain: {-4,-1,2,6} Range: {-2,0,2,4} This is a discrete function - made up of points. State the domain and range: Domain: x -3 Range: y 2 This is a continuous function - made up of a line, ray, segment, or curve. f(x) = x2 Finding the domain and range from the function To find the domain ask if there are any values x cannot be. In this case the answer is No. Therefore, the domain is x x This reads x such that x is an element of the Real Number System. To find the range, ask what kind of numbers you will get when you take a number and square it. The number will always be 0 or positive. Therefore, the range is y y 0 This reads y such that y is greater than or equal to 0. f(x) = x2 Finding the domain and range from the graph Look at the graph and ask what the x-values are on the graph. Since the branches of the parabola extend indefinitely, the values of x extend to negative and positive infinity. Therefore, the domain is x x Look at the graph again and ask what the y-values are. Notice that the lowest yvalue on the graph is 0. Therefore the range is y y 0 f ( x) 1 x Finding the domain and range from the function Domain: Are there any values x cannot be? x cannot be 0 because the square root of 0 = 0 and division by 0 is undefined. x cannot be negative because we cannot take the square root of a negative number. Therefore the domain is xx > 0. Range: What is y when x > 0? y will never be 0 but can be any positive number. Therefore the range is yy > 0. f ( x) 1 x Finding the domain and range from the graph Find the domain. Look at the graph The domain is xx > 0. and ask what are the x-values? Find the range. Look at the graph and ask what are the y-values? The range is yy > 0. Give the domain and range of each function. f(x) = -x4 The domain is The range is x x yy 0 g ( x) 3 2 x The domain is The range is xx > 0 yy > 0 Give the domain and range of the function. f(x) = x + 3 The domain is x x The range is yy 3 p. 108 (16 - 42) Function Notation If there is a correspondence between values of the domain, x, and values of the range, y, that is a function, then y = f(x) read y = “f of x”. (x,y) can be written (x, f(x)) The variable x is called the independent variable and the variable y, or f(x) is called the dependent variable. Evaluating Functions Let f(x) = 3x2 - 2x. Find f(4) To evaluate a function, replace x with the value of the number. f(4) = 3(4)2 - 2(4) = 3(16) - 8 = 40 Let f(x) = 8x - 5. Find f(-2) f(-2) = 8(-2) - 5 = -16 - 5 = -21 Let f(x) = 2x2 + 5x. Find f(3) f(3) = 2(3)2 + 5(3) = 2(9) + 15 = 33 Applying Functions Your pay (P) for working part-time at the Aquarium is a function of the number of hours you work (h). If you make $5.25 an hour, then P(h) = 5.25h. Find your pay for 5 hours of work. Since you worked 5 hours, h = 5. To find your pay, find P(5). P(5) = 5.25(5) = 26.25 Your pay for 5 hours is $26.25 Applying Functions The cost (C) of tickets for the ballet is a function of the number (n) of tickets you buy. If each ticket costs $26.50, then C(n) = 26.5n. Find the cost of 4 tickets. Since you want 4 tickets n = 4. To find the cost, find C(4) C(4) = 26.5(4) = 106. The cost of the tickets is $106. Applying Functions Monthly residential electric charges, c, are determined by adding a fixed fee of $6.00 to the product of the amount of electricity consumed each month, x, in kilowatt-hours and a rate factor of 0.035 cents per kilowatthour. Write a linear function to model the monthly electric charge, c, as a function of the amount of electricity consumed each month, x. c = electricity used + fixed fee c = 0.035x + 6.00 Applying Functions If a household uses 712 kilowatt-hours of electricity in a given month, how much is the monthly electric charge? Evaluate the function for x = 712 c(x) = 0.035x + 6.00 c(712) = 0.035(712) + 6.00 c(712) = 30.92 For 712 kilowatt-hours of electricity, the monthly charge is $30.92 p. 108 (43 – 60, 64 - 69)