# Keystone Problem… Keystone Problem… Set 17 Part 3 © 2007 Herbert I. Gross next.

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Keystone Problem… Keystone Problem… Set 17 Part 3 © 2007 Herbert I. Gross next

You will soon be assigned problems to test whether you have internalized the material in Lesson 17 Part 3 of our algebra course. The Keystone Illustrations below are prototypes of the problems you'll be doing. Work out the problems on your own. Afterwards, study the detailed solutions weve provided. In particular, notice that several different ways are presented that could be used to solve each problem. Instruction for the Keystone Problem © 2007 Herbert I. Gross next

As a teacher/trainer, it is important for you to understand and be able to respond in different ways to the different ways individual students learn. The more ways you are ready to explain a problem, the better the chances are that the students will come to understand. © 2007 Herbert I. Gross next

© 2007 Herbert I. Gross List the members of the set that constitute the graph of f. Keystone Problems for Lesson 17 Part 3 Problem #1a The function f is defined by the rule f(x) = 3x + 1, and its domain is the set A = {0,2,4,6}. next

© 2007 Herbert I. Gross Solution for Problem #1a By definition, the graph of f is the set of all ordered pairs (x,f(x)). In terms of the present exercise… next Therefore, if we let G represent the graph of f, we see that… Domain of f x 3x f(x) = 3x + 1 (x,f(x)) 001(0,1)267(2,7)41213(4,13)61819(6,19) (0,1) (2,7) (4,13) (6,19) (2,7),(4,13), (6,19) G ={ }(0,1),

© 2007 Herbert I. Gross Make sure you understand the difference between the mathematical definition of the graph of a function (that is, as a set of ordered pairs) and the geometric version (which is the set of points that represents the ordered pairs). Note next The geometric version of the graph often gives us an important insight as to how the function behaves. For example, if we locate the members of the graph as points in the plane, we get the points (0,1), (2,7), (4,13) and (6,19), and we can see that these points lie on the same straight line.

More visually… 2468 10 2 16 4 6 8 10 12 0 14 16 18 1218202224 20 (0,1) (2,7) (4,13) (6,19) next © 2007 Herbert I. Gross

This observation verifies what we already know mathematically; namely the fact that f(x) = 3x + 1 tells us that every time x increases by 1 unit, f(x) increases by 3 units. Note next In our geometric graph, this is shown in the equivalent form that when x increases by 2, y increases by 6.

Pictorially, our graph looks like… 2468 10 2 16 4 6 8 10 12 0 14 16 18 1218202224 20 (0,1) (2,7) (4,13) (6,19) next 22 66 © 2007 Herbert I. Gross

Analytic Continuation When we draw the straight line that passes through the given four points, we call the line the analytic continuation of the graph. In this case what we have shown is that if the four given points do lie on the same straight line, the equation of the line has to be… y = 3x + 1. © 2007 Herbert I. Gross next

Analytic Continuation Notice that in this problem, the only information we have is the four given points. Therefore, we drew a dashed line rather than a solid line because seeing only a finite number of points in the plane does not determine what curve the points are on. © 2007 Herbert I. Gross next

© 2007 Herbert I. Gross However, even though the line y = 3x + 1 passes through all four points (0,1), (2,7), (4,13) and (6,19), there are infinitely many other curves that also pass through these four points. One of these other curves (which we shall call C) is shown (in color) on the next slide… Note next

2468 10 2 16 4 6 8 10 12 0 14 16 18 1218202224 20 (0,1) (2,7) (4,13) (6,19) © 2007 Herbert I. Gross C

In other words, when we decide that a finite number of points is enough to determine the curve that passes through them, we are using inductive reasoning. That is, we analyze what we think these points have in common. However, we can never be sure that what we surmised is factually true. Note next

The curve C cannot represent a function. Namely, if C was the graph of a function g, then there could only be one value for, say, g(5). In terms of a graph that means that a vertical line drawn through (5,0) must intersect the curve C at one and only one point. However, notice that there are several points on the curve for which the x-coordinate is 5. Not Every Curve Represents a Function next © 2007 Herbert I. Gross

next © 2007 Herbert I. Gross The x-coordinate of each of the points P, Q, R and S is 5. 2468 10 2 16 4 6 8 10 12 0 14 16 18 1218202224 20 P Q R S Looking at the graph…

next © 2007 Herbert I. Gross 2468 10 2 16 4 6 8 10 12 0 14 16 18 1218202224 20 Looking at the graph… While C doesnt represent a function, it is the union of four curves C 1, C 2, C 3, and C 4 ; each of which does represent a function. C1,C1, C1C1 C2,C2, C2C2 C3,C3, C4;C4; C3C3 C4C4 next

© 2007 Herbert I. Gross 2468 10 2 16 4 6 8 10 12 0 14 16 18 1218202224 20 Looking at the graph… Notice that the line x = 5 intersects each of the curves, C 1, C 2, C 3, and C 4 at one and only one point (e.g., it intersects C 4 only at the point S, etc.) C1C1 C2C2 C3C3 C4C4 next x = 5 S P Q R

© 2007 Herbert I. Gross Not Every Rule is a Function next As an application of what we have just discussed, consider the following… What is incorrect about the following statement? Let f be the function that assigns to every (non-negative) number its square root. next

© 2007 Herbert I. Gross next In order to be a function, f would have to assign one and only one value to each member in its domain. However, as defined in this question, f does not do this. For while it is true that 3 2 = 9, it is also true that ( - 3) 2 = 9. That is, a number such as 9 has two square roots; namely, 3 and - 3.

© 2007 Herbert I. Gross To avoid this ambiguity we can represent f as the union of the two functions g and h, where… Note next g(x) = + ; image of g = {y:y 0}x In other words, g assigns to any non-negative number its non-negative square root. next For example… g(9) = 3 h(y) = - ; image of h = {y:y 0}x

© 2007 Herbert I. Gross And on the other hand… Note next h(x) = - ; image of h = {y:y 0}x means that h assigns to any non-negative number its negative square root. next For example… h(9) = - 3

© 2007 Herbert I. Gross So in terms of g and h, f(x) = g(x) U h(x). next h(x) = - ; image of h = {y:y 0} x In other words, while the rule f is not a function, it is the union of the two functions g and h. next For example… g(x) = + ; image of g = {y:y 0} x f(9) = + 3 = g(9) - 3 = h(9) { = g(9) U h(9). next f(x) = g(x) U h(x).

From a graphing point of view… The curve C 1 is the graph of g, x y and the curve C 2 is the graph of h. next C1C1 C2C2 (9,3) (9, - 3) © 2007 Herbert I. Gross

Graphing the Union … C, which represents the graph of f, is the union of C 1 and C 2. x y next (9, - 3) (9,3) © 2007 Herbert I. Gross

Graphing the Intersection… x y next x = 9 © 2007 Herbert I. Gross P The vertical line x = 9 intersects C at the two points P(9,3) and Q(9, - 3). However, P is only on C 1 (9,3) (9, - 3) Q C 1 (y = + ) x and Q is only on C 2. C 2 (y = - ) x next

© 2007 Herbert I. Gross To avoid confusion we agree to let Note next For example, when we write x mean x +, and we refer to it as the principal square root of x. 4, we mean 2; and if we had wanted to mean - 2, we would have written 4.- If we had meant both 2 and - 2, we would have written ± In this context, when we say the square root of x, we mean the principal square root of x. next

The restriction that a function assigns to each input one and only one output is quite modern. Previously one was allowed to write such things as f(x) =, and refer to f as being a multi-valued function. ±x +x x next © 2007 Herbert I. Gross Historical Note In this context, g(x) =, and h(x) = would have been referred to as single valued branches of f. Nowadays, however, function means single-valued function. next

© 2007 Herbert I. Gross Keystone Problems for Lesson 17 Part 3 The function f, mentioned in Problem 1b, is defined by the rule f(x) = 3x + 1 Problem #1b Suppose the line L is the analytic continuation of the geometric graph of f. Is the point (20,61) on the line L?

next © 2007 Herbert I. Gross Solution for Problem #1b A point (x,y) is on this line L if and only if… next So to see whether (20,61) is on this line, we must replace x by 20 and y by 61 in our equation and see if we obtain a true statement. Doing this we see that… 61 = 3(20) + 1 y= 3x + 1 = 60 + 1= 61 Thus (20,61) is a point on this line.

© 2007 Herbert I. Gross Algebraic & Geometric Relationships next Notice that the beauty of knowing the algebraic relationship between the x and y coordinates of the points on the line eliminates the need for us to have to draw the line. In short, one of Descartes contributions is that it allows us to view geometric shapes algebraically and algebraic equations geometrically.

© 2007 Herbert I. Gross The fact that the slope of L is 3 and that L passes through the point (6,19), gives us another way to solve the problem. Note next However, knowing that the equation of the line is y = 3x + 1, we can determine whether a point belongs to the line with a minimum of effort. Namely, in going from (6, 19) to (20, f(x)), x increases by 14. Hence, y increases by 3 × 14, or 42. Since 19 + 42 = 61, we see that (20, 61) is on the line.

Knowing that the equation of the line is y = 3x + 1 tells us much more than that (20,61) is on the line. © 2007 Herbert I. Gross More generally, all points (20, y) for which y < 61 lie below the line. Note next For example, the fact that the point (20,60) is below the point (20,61) means that (20,60) is below the line y = 3x + 1.

In a similar way, if y > 61, the point (20, y) lies above the line y = 3x + 1. In other words, all points (20,y) for which y > 61 lie above the line y = 3x + 1. Of course, there was nothing special about our choice of x = 20. © 2007 Herbert I. Gross next Thus, we may generalize this discussion in the following slide… Note

The line whose equation is y = 3x + 1 divides the xy-plane into two half planes, namely… next If y < 3x + 1, the point (x,y) is below the line. That is, {(x,y): y < 3x + 1} is the half plane that lies below the line y = 3x + 1 If y > 3x + 1, the point (x,y) is above the line. That is, {(x,y): y < 3x + 1} is the half plane that lies above the line y = 3x + 1 © 2007 Herbert I. Gross next Note

y = 3x + 1 y < 3x + 1 y > 3x + 1 next © 2007 Herbert I. Gross Pictorially…

Finally, the same discussion applies to any line when written in the form y = mx + b. next If y = mx + b, the point (x,y) is on the line. If y < mx + b, the point (x,y) is below the line. If y > mx + b, the point (x,y) is above the line. © 2007 Herbert I. Gross Namely… next Note