MAT 12 - SEC 021 PRECALCULUS SUMMER SESSION II 2014 LECTURE 3

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1 MAT 12 - SEC 021 PRECALCULUS SUMMER SESSION II 2014 LECTURE 3 JAMIE HADDOCK 1. Agenda Functions Composition Graphs Average Rate of Change Functions Definition 1. Let A and B be two nonempty sets. A function, f, from A to B is a rule of correspondence that assigns to each element in set A exactly one element in B. In this course, A and B will always be sets of real numbers, although they will not always be all real numbers. The set A is called the domain of the function, while B is the codomain and contains the range of the function. Definition 2. The domain of the function, f is the set of real numbers the function f is defined upon. Definition 3. The range of the function, f is the set of real numbers that can be reached by the function. For every number, r in the range, R, there is a number d in the domain, D so that f(d) = r. In general, the letter in a function that represents the elements from the domain is called the independent variable. The letter representing elements of the range is called the dependent variable. Example 1. Consider the function y = 1 x+3. First, note that the independent variable is x, while the dependent variable is y. Now, the domain is all real numbers on which the function is defined. Now, the only number which poses any problems for us is x = 3, since then the function is undefined (division by 0). Thus, the domain of the function is (, 3) ( 3, ). Now, to find the range, we solve for x for all numbers in the domain (i.e. x 3). y = 1 x + 3 y(x + 3) = 1 (we can do this since (x + 3) 0) xy + 3y = 1 xy = 1 3y x = 1 3y y Now, this tells us that there is an x that achieves y under the function for every y except for 0, since then x would have to be. Thus, the range is (, 0) (0, ). Date: August 8th,

2 MAT 12 - SEC 021 PRECALCULUS SUMMER SESSION II 2014 LECTURE 3 2 Often, we will use function notation (representing a function by a letter) as in f(x) = x 2. This says that the function, f has independent variable x and is computed by squaring x. x+2 Question 1. Determine the domain and range of the function f(x) = x 1.

3 MAT 12 - SEC 021 PRECALCULUS SUMMER SESSION II 2014 LECTURE 3 3 Example 2. (Two types of demand functions) Suppose a company sells ticks and tacks. The company has been selling them for a long time and has found out that if they want to sell n pounds of ticks, they must price them at p(n) = 19.01n cents. They also know if they price their tacks at p cents, they will sell d(p) = p. Suppose the company is making the same amount from selling ticks as they are from selling tacks and they sell 1100 tacks. How many pounds of ticks did they sell and what were the ticks and tacks priced? First, let s find out how much the tacks cost: d(p) = p = =.5p 712 = p Thus, their tacks cost 712 cents or $7.12 each. Now, how much did the company earn from selling tacks? d(p) p = = The company made $7832 from the tacks. Now we can figure out how much the ticks cost and how many the company sold. However, we actually have to do this at the same time since we know that they made $7832 from them = p(n) n Now, we can use the quadratic formula to solve this: = (19.01n) n = 19n.01n 2 0 =.01n n =.01n 2 19n n = 19 ± 361 4(.01)(7832) 19 ± = (we re assuming they sell the smaller of the two numbers) Thus, they sell pounds of ticks priced at p(1209.2) = 19.01(1209.2) = =

4 MAT 12 - SEC 021 PRECALCULUS SUMMER SESSION II 2014 LECTURE 3 4 x 2 x+1 equal to 10? Question 2. For what x is f(x) = Composition Definition 4. If f and g are two functions then (1) (f + g)(x) = f(x) + g(x) (2) (f g)(x) = f(x) g(x) (3) (fg)(x) = f(x)g(x) (4) are all functions. (f/g)(x) = f(x)/g(x) if g(x) 0 Definition 5. Given two functions f and g, the composition of f and g is the function f g defined by (f g)(x) = f[g(x)] The domain of f g consists of those inputs x in the domain of g for which g(x) is in the domain of f. Example 3. Suppose f(x) = x and g(x) = x 2. Compute f g and g f and find their domains and ranges. First, f g = f[g(x)] = f(x 2) = x 2. Now, g f = g[f(x)] = g( x) = x 2. Now, when x = 2, we get (f g)(2) = 2 2 = 0 and (g f)(x) = Now we can see that these functions are not the same. Now, the domain of (f g)(x) = x 2: We can t take the square root of a negative number, so x 2 0 and x 2. The range of (f g)(x) will be the same as the range of f(x) = x, which is [0, ). Now, the domain of (g f)(x) = x 2 is all numbers for which x is defined: x 0. The range is the range of x but shifted down. The range of x is [0, ) so the range of (g f)(x) is [ 2, ).

5 MAT 12 - SEC 021 PRECALCULUS SUMMER SESSION II 2014 LECTURE 3 5 Question 3. Suppose you know that the radius of a spherical bubble is given by r(t) =.15t inches where t is in seconds. How is the volume of the bubble changing with time? What is the volume of the bubble after 3 seconds? Graphs Definition 6. The graph of a function f in the x y plane consists of those points (x, y) such that x is in the domain of f and y = f(x). Question 4. Sketch the graph of the function f(x) = x. This example illustrates a way we can check if a graph represents a function: Vertical Line Test A graph in the x y plane represents a function of x provided that any vertical line intersects the graph in at most one point. Question 5. Use the graph of y 2 + x 2 = 1 to determine if this equality represents a function in x. What are functions that satisfies this equality? Sketch the graphs of these functions.

6 MAT 12 - SEC 021 PRECALCULUS SUMMER SESSION II 2014 LECTURE 3 6 Example 4. What are the domain and range of the function depicted in below? The domain is all x values for which the function is defined: (, 2) [3, 7). The range is all y values which the function achieves: ( 1, 2) [7, ). What is f(4)? What is f(7)? Question 6. Graph h(x) = x. 1 + x x < 1 Example 5. Graph c(x) = x 2 1 x 1. x x > 1 Question 7. Graph g(t) = { t t < 0 t t 0.

7 MAT 12 - SEC 021 PRECALCULUS SUMMER SESSION II 2014 LECTURE 3 7 Question 8. Use the graphs of f(x) = x 2 and g(x) = x to find where x 2 = x Average Rate of Change Definition 7. The number f(x 0 ) is the maximum value of a function f if the inequality f(x 0 ) f(x) holds for every x in the domain of f. Question 9. Draw a function with a maximum and write the point (x 0, f(x 0 )). Definition 8. The number f(x 0 ) is the minimum value of a function f if the inequality f(x 0 ) f(x) holds for every x in the domain of f. Question 10. Draw a function with a minimum and write the point (x 0, f(x 0 )).

8 MAT 12 - SEC 021 PRECALCULUS SUMMER SESSION II 2014 LECTURE 3 8 Definition 9. A function f is increasing on an interval provided for all pairs of numbers a and b in the interval, if a < b then f(a) < f(b). Question 11. Draw a function that is increasing on the interval (0, 2). Definition 10. A function f is decreasing on an interval provided for all pairs of numbers a and b in the interval, if a < b then f(a) > f(b). Question 12. Draw a function that is decreasing on the interval [ 1, 0]. Definition 11. The average rate of change of a function f on the interval [a, b] is the slope of the line joining (a, f(a)) and (b, f(b)) which is f(b) f(a). b a Question 13. What is the average rate of change of the function f(x) = x 2 on the interval [0, 3], [0, 2] and [0, 1]? Can you generalize this to find what the average rate of change is on the interval [0, b]?

9 MAT 12 - SEC 021 PRECALCULUS SUMMER SESSION II 2014 LECTURE 3 9 Example 6. What is the average rate of change of the function f(x) = x 2 on the interval [x, x + h]? f(x + h) f(x) x + h x = (x + h)2 x 2 h = x2 + 2xh + h 2 x 2 h 2xh + h2 = h = 2x + h Question 14. What is the average rate of change of the function f(x) = 2x 2 + x on the interval [x, b]?