1.3 Representations of Functions 31 Specific Example General Statement Symbolic Representation P is a if and only if P is a rectangle with square sides of equal length A if and only if BA⇔B P is a if P is a rectangle with A if B square sides of equal length equivalently equivalently If P is a rectangle with then P is a if B then AB⇒A sides of equal length square P is a only if P is a rectangle with A only if B square sides of equal length equivalently equivalently If P is then P is a rectangle with if A then BA⇒B asquare sides of equal length The graphs of some functions are given in the figure below. V f(V ) V π (a) f(V) = A(r) (r, A(r)) r A 9 2 (b) A(r)= πr V C(V) volume height (c) the calibration function C(V) for the evaporating flask Figure 1.16 Graphs of mappings Q, R, and S from Exercise 1.1 are given in Figure 1.17. You should now be able to check visually that Q and R are functions while S is not a function. x S 13 6 13 6 x Q (3, Q(3)) = (3, 6) R x 13 6 6 4 2 –6 –4 –2 (3, R(3)) = (3, ) 16 15 14 13 12 11 π π (a) (b) (c) Figure 1.17 32 CHAPTER 1 Functions Are Lurking Everywhere EXERCISE 1.3 Two of the four graphs given in Figure 18 are the graphs of functions. 15 Which two are they? Can you come up with a rule for determining whether or not a given graph is the graph of a function? x y y x x y y x (a) (b) (c) (d) Figure 1.18 Answer We can tell that the relationships represented in Figures 1.16(a)–(c), 1.17(a) and (b), and 1.18(b) and (d) are, in fact, functions. The test for a function is that every input must have only one output assigned to it; graphically, this means if we draw a vertical line through any point on the horizontal axis, this line cannot cross the graph in more than one place. (The vertical line will not cross the graph at all if it is drawn through a point on the horizontal axis that is not a valid input, i.e., not in the domain of the function.) This test for deciding if a graph represents a function is called the vertical line test. x y y x x y y x (a) (b) (c) (d) Not the graph of a function Not the graph of a function Figure 1.19 Vertical line test Let’s return to the problem of calibrating bottles. When we calibrate a bottle we put in a known volume of liquid and the calibration function C(v) produces a height. Once we have calibrated a bottle we can then use it as a measuring device. We turn the procedure around so that the height becomes the input while the volume is the output. The key ingredient that makes the original volume-height assignment useful is that not only does each volume correspond to one height but each height corresponds to only one volume. The calibration function is 1-to-1. A function f that is 1-to-1 has an inverse function that “undoes” f . The domain of the inverse function is the range of f ;iff maps a to b then its inverse function mapsbtoa. 16 EXERCISE 1.4 Given the graph of a function, how can we determine whether the function is 1-to-1? 15 The conventions about hollow and filled circles are the same as they are for interval notation. A hollow circle denotes a point that is not on the graph, while a filled circle indicates that the point is on the graph. 16 Sometimes people mistakenly think of the words “inverse” and “reciprocal” as being the same. The inverse of the function f undoes f .Iff adds 3 to its input then its inverse function subtracts 3; if f multiplies its input by 3 then its inverse function divides by 3. If f maps 7 to 11 then its inverse function maps 11 to 7. On the other hand, the reciprocal of N is simply 1/N . 1.3 Representations of Functions 33 Answer By the horizontal line test: If any horizontal line intersects the graph in more than one place, then the function is not 1-to-1; if no horizontal line intersects the graph in more than one place, then the function is 1-to-1. Functions: The Grand Scheme In this text we will be looking at functions of one variable—but not all functions are functions of one variable. For instance, suppose you were to calculate the value, V ,of the change in your pocket. V is a function of q, d, n, and p, where q, d, n, and p are the number of quarters, dimes, nickels, and pennies, respectively, in your pocket. We can write this function as V = f(q,d,n,p) = 25q + 10d + 5n + p. As another example, suppose we denote by V the volume of a fixed mass of gas, by T its temperature on the Kelvin scale, and by P its pressure. Then the combined gas law tells us that PV T = k, where k is a constant, or, equivalently, V = k P · T . We see that the volume of the gas will depend upon both the pressure and the temperature. V is a function of two variables, V = f(P,T).Ifpressure is held constant, then volume is directly proportional to temperature; as the temperature of a gas goes up, the volume goes up as well. V = (a constant) · T is the gas law of Charles and Gay-Lussac. In other words, if pressure is held constant, then V can be expressed as a function of one variable. Think back to Example 1.9 where we were expressing the volume of water in a conical coffee filter as a function of the height of the water. We began by writing V = 1 3 πr 2 h, where both r, the radius of the liquid, and h, the height of the liquid are variables. We wanted to express V as a function of only h. In this example we cannot simply say “let r be a constant,” since h cannot vary without r varying. Instead, we found a relationship between h and r that allowed us to express the former in terms of the latter, resulting in a function of only one variable. Because our focus will be on functions of one variable, when we do modeling we need to determine one independent variable and one dependent variable, for a total of two variables. If we appear to have more variables, then it is necessary to do one of two things: Figure out how we can express one of the variables in terms of another, or be less ambitious with our model, treating some quantities as constants when it is reasonable to do so. 34 CHAPTER 1 Functions Are Lurking Everywhere PROBLEMS FOR SECTION 1.3 1. (a) Consider the following graphs. For each graph decide whether or not y is a function of x. If it is a function, determine the range and domain. x y -5 -4 -3 -2 -1 123456 5 4 3 2 1 -1 -2 x y -3 -2 -1 1 2 3 456 5 4 3 2 1 -1 -2 -3 x y -4 -3 -2 -1 123456 5 4 3 2 1 -1 -2 7 -3 -4 x y -5 -4 -3 -2 -1 123456 3 2 1 -1 -2 x y -5 -4 -3 -2 -1 123456 3 2 1 -1 -2 x y -5 -4 -3 -2 -1 123456 3 2 1 -1 -2 x y -5 -4 -3 -2 -1 1 2 3 4 5 6 5 4 3 2 1 -1 -2 x y -5 -4 -3 -2 -1 123456 5 4 3 2 1 -1 -2 x y -5 -4 -3 -2 -1 123456 5 4 3 2 1 -1 -2 (a) (b) (c) (d) (e) (f) (g) (h) (i) (b) Are any of the functions from part (a) 1-to-1? If so, which ones? In Problems 2 through 4 find the domain of each of the functions. 2. (a) f(x)= 1 x+2 (b) g(x) = 5 x 2 +4 1.3 Representations of Functions 35 3. (a) f(x)= √ x (b) g(x) = √ x − 3 (c) h(x) = √ x 2 − 4 4. (a) f(x)=  1 x+1 (b) g(x) =  x x+1 (Hint: Keep in mind that if the numerator and denominator of a fraction are both negative, then the fraction is positive.) 5. Find the range of the functions in Problems 2 through 4. 6. Let g(x) be the function graphed below with domain [−5, 4]. Use the graph to answer the following questions. (a) What is the range of g? (b) Is g(x) 1-to-1? (c) Where does g take on its highest value? Its lowest value? (d) What is the highest value of g? (e) What is the lowest value of g? (f) For what x is g(x) =0? (g) Approximate g(0). (h) Approximate g(−4). (i) Approximate g(π) and g(−π). (j) For approximately what values of x is g(x) = 1? (k) For approximately what values of x is g(x) < −1? x 4321 1 – 1 – 2 2 3 – 1 – 2 – 3 – 4 – 5 ( – 5, – 2) ( – 3, 3) (4, – 1) g 7. Two functions are equivalent if they have the same domain and the same input/output relationship. The first function listed on each line below is called f . Which of the functions listed on each line are equivalent to f ? The domain of each function is the set of all real numbers. (Be careful to think about the sign of each function.) (a) f(x)=−2x 2 g(w) =(−2w) 2 i(t) =2(−t 2 ) j(x) =− √ 2x 2 √ x 2 √ 2 (b) f(x)=(2x −1) 2 g(c) =(1 −2c) 2 h(t) =1 − 2t 2 j(x)=1−(2x) 2 (c) f(x)= √ x 2 λ(m) =m T (x) =  (−x) 2 ϕ(s) =|s| 36 CHAPTER 1 Functions Are Lurking Everywhere 8. Below is a graph of the function f . Use it to approximate the following. (a) The value of f at x =−1, x = 0, and x = 1 (b) All x such that f(x)=0 (c) All x such that f(x)=2 (d) −f(0)+2f(3) x f –2 –1 –1 1 1 2 3 234 9. Below is a graph of the function g. Use it to approximate the following. (a) g(−2), g(0), g(1) (b) All t such that g(t) = 0 (c) All t such that g(t) = 1 (d) g(−1) + g(1) 2 (e) g(3) + 3g(1) t g –2–3 –1 –1 –2 1 1 2 2345 10. The graph of h is given at the top of page 37. (a) Give the domain and range of h. (b) Evaluate h(0), h(1), h(2). (c) Fill in the output values on the table. 1.3 Representations of Functions 37 x h(x) −0.5 −0.01 0 0.1 0.5 1.5 1.95 2 2.01 (d) Find all x such that h(x) = 3. Use inequalities in your answer. (e) Find all x such that h(x) = 1. Use interval notation, paying attention to [ versus (. x h –2–3 –1 1 1 2 3 4 234 11. The graph of k is given below. (a) Give the domain and range of k. (b) Find 2k(−1) + [k(−1)] 2 + k((−1) 2 ). (c) Find all t such that k(t) = 2. x k 1 –1 1 2 3 (2, 1) (–1, 2) (3, 3) (4, 2) 234 12. Consider the functions f , g, h, and k presented as graphs in Problems 8 through 11. Which, if any, of these functions are 1-to-1? 38 CHAPTER 1 Functions Are Lurking Everywhere 13. The graphs of f and g are given below. Approximate the following. (a) All x such that f(x)= g(x) (b) All x such that g(x)>f(x) x y g f 1–1–2–4 2 3 4 5 6 7 8 14. Determine whether or not the given graph is the graph of a function. x y x y x y x y (a)(b) x y x y (d)(d) (e) (f) x y (g) 15. Draw the graph of a function f that is 1-to-1 and a function g that is not 1-to-1. 1.3 Representations of Functions 39 x f (– 4, –3) (5, 3) x k (–3, –2) (4, 6) x g (3, 2) (– 4, –3) The graph of a function is given in Problems 16 through 22. Determine the range and domain. Is the function 1-to-1? 16. 17. x h (–3, –1) 1 (4, 5) 18. 19. x m 3–3 –2 2 20. 21. x j (–5, – 4) (4, 5) 22. x (–3, .01) (3, –.01) ഞ 40 CHAPTER 1 Functions Are Lurking Everywhere 23. You make yourself a hot cup of tea. The telephone rings, distracts you, and when you return the tea is at room temperature. Sketch a rough graph of the function T(t),where t is the time, in minutes, from when the phone rings, and T(t)is the temperature of the tea at time t. 24. Your neighbor prunes his hedge once every three weeks. Sketch a rough graph of h(t), the height of the hedge as a function of time, where t = 0 corresponds to a pruning session, and t is measured in weeks. Let the domain of the function be [0, 10]. 25. First Slice is a “come from behind” horse. He starts out slowly and comes on at the end. Tass is a sprinter, starting out fresh and tiring at the end. First Slice and Tass both run a mile in 1:12. On the same set of axes, graph F(t)and T(t),the distances traveled by First Slice and Tass, respectively, as a function of time t, t in seconds. 26. Express the area of a circle as a function of: (a) its diameter, d. (b) its circumference, c. 27. Express the surface area of a cube as a function of the length s of one side. 28. A closed rectangular box has a square base. Let s denote the length of the sides of the base and let h denote the height of the box, s and h in inches. (a) Express the volume of the box in terms of s and h. (b) Express the surface area of the box in terms of s and h. (c) If the volume of the box is 120 cubic inches, express the surface area of the box as a function of s. 29. You are constructing a closed rectangular box with a square base and a volume of 200 cubic inches. If the material for the base and lid costs 10 cents per square inch and the material for the sides costs 7 cents per square inch, express the cost of material for the box as a function of s, the length of the side of the base. 30. A rectangle is inscribed in a semicircle of radius R, where R is constant. (a) Express the area of the rectangle as a function of the height, h, of the rectangle, A = f (h). (b) Express the perimeter of the rectangle as a function of the height, h, of the rectangle, P = g(h). R h . rectangular box has a square base. Let s denote the length of the sides of the base and let h denote the height of the box, s and h in inches. (a) Express the volume of the box in terms of s and. fresh and tiring at the end. First Slice and Tass both run a mile in 1:12. On the same set of axes, graph F(t )and T(t),the distances traveled by First Slice and Tass, respectively, as a function of. the functions f , g, h, and k presented as graphs in Problems 8 through 11. Which, if any, of these functions are 1 -to- 1? 38 CHAPTER 1 Functions Are Lurking Everywhere 13. The graphs of f and