Section 1.1

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Section 1.1: Functions and Models On Line MAT 107 Objectives: Explain the term “function” Recognize and use the 5 different methods for specifying a function Find the domain and range of a function defined by an equation Use the Vertical Line Test to determine whether a graph represents a function. Explain the term “mathematical model”

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“A function is a rule of correspondence that assigns to each element of one set exactly one element of a second set.” Survey Questions What is your age? What is your shoe size? What is the name of your brother or sister? This is a function. Each element of the first set is assigned to exactly one element of the second set.

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“A function is a rule of correspondence that assigns to each element of one set exactly one element of a second set.” Survey Questions What is your age? Tabitha Bill Cameron Suzanne 25 40 51 17 32 This is a function. Each element of the first set is assigned to exactly one element of the second set.

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“A function is a rule of correspondence that assigns to each element of one set exactly one element of a second set.” Survey Questions This is a function. Each element of the first set is assigned to exactly one element of the second set. What is your shoe size?

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“A function is a rule of correspondence that assigns to each element of one set exactly one element of a second set.” Survey Questions This is not a function. Cameron is assigned to more than one element of the second set. What is the name of your brother or sister? Cameron Cameron

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“A function is a rule of correspondence that assigns to each element of one set exactly one element of a second set.” Survey Questions How many brothers and sisters do you have? This is a function. Each element of the first set is assigned to exactly one element of the second set.

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“The function may be defined by …” 1) 2) 3) 4) 5)

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“The function may be defined by …” 1) a set of ordered pairs (-1,3)

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“The function may be defined by …” 1) a set of ordered pairs 2) a table 0 is assigned to 1 2 is assigned to 3 3 is assigned to 7 -1 0 2 3 3 1 3 7

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“The function may be defined by …” 1) a set of ordered pairs (-1,3) , (0,1), (2,3), (3,7) 2) a table 3) a graph (-1,3) (0,1) (2,3) (3,7) x y

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“The function may be defined by …” 1) a set of ordered pairs (-1,3) , (0,1), (2,3), (3,7) 2) a table 3) a graph 4) an equation f(x) = x2 –x + 1

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“The function may be defined by …” (-1,3) , (0,1), (2,3), (3,7)

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“The function may be defined by …” 5) a verbal expression Examples: Survey results for the question “What is your age?” The prices of items at the grocery score The value of the Dow Jones Industrial Average (1950-2010) The monthly car payment required if the entire price ($35,000) is financed at 8% for t years. Average income as a function of the number of years of education The number of violent crimes reported by ZIP code

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I recommend that you take a short break here, before continuing on to the material on domain and range. You can click the pause button and minimize this browser window, or you can close this window and fast forward to the same spot later. For the next topic you will need your graphing calculator.

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has range R = { 1, 3, 7 } The domain of a function is the set of all allowed input values. The range of a function is the set of all realized output values. has domain D = { -1, 0, 2, 3 }

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It appears that there are some y-values that can’t occur. If we zoom in on the graph, we will eventually deduce that y=3/4 is the lowest value. The domain of a function is the set of all allowed input values. The range of a function is the set of all realized output values.

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Finding the domain of a function defined by an equation Start with the domain as the set of all real numbers If the equation has a denominator, exclude any numbers that give a zero denominator. If the equation has a radical of even index, exclude any numbers that cause the expression inside the radical to be negative. The numbers x=-1 and x=1 give a zero denominator. The radical has even index 2. Domain D = all real numbers except -1 and +1 = ( -, -1)  (-1, 1)  (1, ) 2x – 4 < 0 occurs when x < 2 Domain D = all real numbers except x<2 = [2, )

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Finding the range of a function defined by an equation Graph the function on your calculator Decide visually which values are excluded from the range Zoom in if necessary to determine precise locations Range R = all real numbers except 0<y<2 = ( -, 0]  (2, ) If we look more carefully, we see that y=0 can occur, but y=2 cannot occur.

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Finding the range of a function defined by an equation Graph the function on your calculator Decide visually which values are excluded from the range Zoom in if necessary to determine precise locations We know that the domain is [2,). So we can evaluate f(2) = 3 + 01/2 = 3 to see that y=3 occurs. Range R = all real numbers except y<3 = [3, ) It might not be possible to determine from “trace” whether the value y=3 occurs or not.

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Vertical line test: “A set of points in a coordinate plane is the graph of a function if and only if no vertical line intersects the graph in more than one point.” Two parts Examples FAIL – not a function FAIL – not a function PASS – it is a function If a vertical line hits the graph twice, it is not a function If all vertical lines hit once or not-at-all, then it is a function “It FAILS the test.” “It PASSES test.”

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“Modeling is the process of translating real-world information into a mathematical form so that it can be applied and then interpreted in the real-world setting.” 1. Translate into mathematical form 3. Apply and interpret in real world Step 2 is learning how to manipulate, describe and solve things in the math world.

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“A Mathematical Model is a functional relationship (usually in the form of an equation) that includes not only the function rule but also descriptions of all involved variables and their units of measure.” Here is an equation: Floss = C  Q2  L It probably describes something, but without descriptions and units for the variables it’s pretty worthless. Here is a model: (from www.FirefighterNation.com) The frictional pressure loss Floss for a fire hose in PSI is given by the formula Floss = C  Q2  L where C is the hose coefficient, Q is the water flow rate in GPM divided by 100, and L is the length of the hose in feet divided by 100. It assumes no appliances or elevation change. The model contains the information needed to allow the formula to be connected to a real-world situation. This important model is used to make decisions about ordering equipment or for substituting equipment to avoid mismatches of pressure tolerance.

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Section 1.1: Functions and Models On Line MAT 107 Objectives: Explain the term “function” Recognize and use the 5 different methods for specifying a function Find the domain and range of a function defined by an equation Use the Vertical Line Test to determine whether a graph represents a function. Explain the term “mathematical model”