function relations and domain of functions

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1 function relations and domain of functions Module 3 : Investigation 2 MAT 170 Precalculus September 7, 2016

2 what is a function? Recall from last time, that a function consists of 3 parts : (1) The set of input values, which we call the domain. (2) The set of output values, which we call the range. (3) A rule that assigns each input value in the domain to exactly one output value. 2

3 how we can express a function We can express the rule of a function in four different ways : (1) In words - Let f be the function that takes an input of a real number and outputs the square of the input. (2) In a table of values - x f(x)

4 how we can express a function We can express the rule of a function in four different ways : (3) By a graph - (4) By a formula - 10 f(x) = x

5 question 3 Pat was driving his car across Kansas with the cruise control on, when his gas gauge broke. At the moment the gauge broke, he had 15 gallons of gas in the car s gas tank and his gas mileage was 42 miles per gallon (assume he maintains this gas mileage by leaving the cruise control on). (a) How many gallons does pat have left after he has driven 84 miles? 150 miles? (b) Define a function f to determine the number of gallons left in Pat s tank f(x) in terms of the number of miles driven since the gas gauge broke x. (c) What is the domain of f? (d) What is the range of f? 5

6 question 3 - solutions (a) How many gallons does pat have left after he has driven 84 miles? 150 miles? At 84 miles Pat will have used = 2 gallons. This means he will have 15 2 = 13 remaining. By a similar argument he will have approximately 11.4 gallons remaining after driving 150 miles. (b) Define a function f to determine the number of gallons left in Pat s tank f(x) in terms of the number of miles driven since the gas gauge broke x. f(x) = 15 x 42 6

7 question 3 - solutions (c) What is the domain of f? Ask yourself what restrictions there are on the values that can be used as inputs x for f. We know that x must be non-negative (Pat cannot drive negative miles), so x 0. Since Pat cannot continue driving after he has run out of gas, we know that x a, where a is the number of miles at which Pat runs out of gas. To determine a, we solve 15 a 42 = 0 15 = a = a. Therefore, Domain : 0 x 630 or [0, 630]. 7

8 question 3 - solutions (d) What is the range of f? Ask yourself what are all of the possible outputs f(x) for f. The tank starts with 15 gallons when x = 0 miles, and ends with 0 gallons when x = 603 miles. Since every number of gallons between 0 and 15 will occur : Range : 0 f(x) 15 or [0, 15]. 8

9 question 3 Pat was driving his car across Kansas with the cruise control on, when his gas gauge broke. At the moment the gauge broke, he had 15 gallons of gas in the car s gas tank and his gas mileage was 42 miles per gallon (assume he maintains this gas mileage by leaving the cruise control on). (e) Let f be as in part (b). What does f(100) represent in this situation? (f) What does x represent in this situation? 42 (g) What does 15 x represent in this situation? 42 (h) What are the maximum and minimum values that f(x) can assume in the context of this situation? 9

10 question 3 - solutions (e) Let f be as in part (b). What does f(100) represent in this situation? The number of gallons left in Pat s tank after driving 100 miles. (f) What does x represent in this situation? 42 The number of gallons used by Pat to drive x miles. (g) What does 15 x represent in this situation? 42 The remaining gallons of gas in Pat s car after driving x miles. (h) What are the maximum and minimum values that f(x) can assume in the context of this situation? Maximum : 15 Minimum : 0 10

11 question 3 Pat was driving his car across Kansas with the cruise control on, when his gas gauge broke. At the moment the gauge broke, he had 15 gallons of gas in the car s gas tank and his gas mileage was 42 miles per gallon (assume he maintains this gas mileage by leaving the cruise control on). (e1) Let f be as before. Construct a graph of f with the input quantity (the number of miles driven since the gas gauge broke) on the horizontal axis, and the output quantity on the vertical axis. (e2) Explain what the graph of f conveys about the relationship between the number of miles Pat has driven since his gas gauge broke x, and the number of gallons of gas left in Pat s tank f(x). (j) What does the point (0, 15) on the graph represent in the context of this situation. 11

12 question 3 - solutions 50 (e1) Let f be as before. Construct a graph of f with the input quantity (the number of miles driven since the gas gauge broke) on the 40 horizontal axis, and the output quantity on the vertical axis

13 question 3 - solutions (e2) Explain what the graph of f conveys about the relationship between the number of miles Pat has driven since his gas gauge broke x, and the number of gallons of gas left in Pat s tank f(x). The value of f(x) changes at a constant rate with respect to x. (j) What does the point (0, 15) on the graph represent in the context of this situation. When Pat has driven 0 miles (x = 0), the number of gallons of gas in his tank is 15 (f(0) = 15). 13

14 vertical and horizontal intercepts When considering the graph of a function, there are two types of points that tend to be a particular interest. Definition The point where the graph of a function f crosses the vertical axis is called the vertical intercept of f. The points where the graph of a function f crosses the horizontal axis are called the horizontal intercepts of f. The vertical intercept (if it exists) is the point (0, b) where f(0) = b. The horizontal intercepts (if they exist) are points of the form (a, 0) where f(a) = 0. Note that there is at most one vertical intercept, while there may be more than one horizontal intercept. Why? 14

15 vertical and horizontal intercepts y=f (x)

16 vertical and horizontal intercepts y=f (x) Vertical intercept

17 vertical and horizontal intercepts y=f (x) Vertical intercept Horizontal intercepts

18 question 3 Let f be the function giving the number of gallons left in Pat s tank f(x), in terms of the number of miles driven since the gas gauge broke. (k) Since the vertical intercept of the function f occurs where x = 0, what general method can you use to determine the vertical intercept of the graph of a function f. (l) What is the value of x when f(x) = 0? What point of the graph of f corresponds to f(x) = 0? What does this point convey given the situation? (m) Since the horizontal intercept of a function f occurs where f(x) = 0, what general method can you use to determine the horizontal intercept of the graph of a function f? 18

19 question 3 - solutions (k) Since the vertical intercept of the function f occurs where x = 0, what general method can you use to determine the vertical intercept of the graph of a function f. Input 0 into the function f to determine the output f(0). The vertical intercept will then be the point (0, f(0)). (l) What is the value of x when f(x) = 0? What point of the graph of f corresponds to f(x) = 0? What does this point convey given the situation? We determined that f(x) = 0 when x = 630. The point on the graph of f corresponding to f(x) = 0 is the horizontal intercept (630, 0). 19

20 question 3 - solutions (m) Since the horizontal intercept of a function f occurs where f(x) = 0, what general method can you use to determine the horizontal intercept of the graph of a function f? Solve the equation f(x) = 0 for x. Suppose the solution is x = a. The horizontal intercept will then be (a, 0). 20

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Activity A: Sidewalk Replacement.

Activity A: Sidewalk Replacement. The following activity is from Introductory Algebra and focuses and synthesizes reading a graph with writing an equation. The piecewise function is typical in real world