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

Keystone Problem…Keystone Problem…

Set 17 Part 3

© 2007 Herbert I. Gross

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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 we’ve provided. In particular, notice that several different ways are presented that could be used to solve each problem.

Instruction for the Keystone Problem


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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

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List the members of the set that constitute the graph of f.

Keystone Problems for Keystone Problems for Lesson 17 Part 3Lesson 17 Part 3

Problem #1aProblem #1a

The function f is defined by the rule f(x) = 3x + 1, and its domain is

the set A = {0,2,4,6}.

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Solution for Problem #1aSolution for Problem #1aBy definition, the graph of f is the set of all ordered pairs (x,f(x)). In terms of the present exercise…

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Therefore, if we let G represent the graph of f, we see that…

Domain of fx

3xf(x)

= 3x + 1(x,f(x))

0 0 1 (0,1)

2 6 7 (2,7)

4 12 13 (4,13)

6 18 19 (6,19)

(0,1)(0,1)

(2,7)(2,7)

(4,13)(4,13)

(6,19)(6,19)

(2,7),(4,13),(6,19)G ={ }(0,1),


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).

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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.

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More visually…

2 4 6 8 10

2

16

4

6

8

10

12

0 14

14

16

18

12 18 20 22 24

20

(0,1)

(2,7)

(4,13)

(6,19)

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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

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In our geometric graph, this is shown in the equivalent form that when x increases by 2,

y increases by 6.

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Pictorially, our graph looks like…

2 4 6 8 10

2

16

4

6

8

10

12

0 14

14

16

18

12 18 20 22 24

20

(0,1)

(2,7)

(4,13)

(6,19)

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22

66


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

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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.


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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

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2 4 6 8 10

2

16

4

6

8

10

12

0 14

14

16

18

12 18 20 22 24

20

(0,1)

(2,7)

(4,13)

(6,19)


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.

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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 Not Every Curve Represents a FunctionRepresents a Function

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The x-coordinate of each of the points P,

Q, R and S is 5.

2 4 6 8 10

2

16

4

6

8

10

12

0 14

14

16

18

12 18 20 22 24

20

P

Q

R

S

Looking at the graph…

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2 4 6 8 10

2

16

4

6

8

10

12

0 14

14

16

18

12 18 20 22 24

20


While C doesn’t represent a function, it is the union of four curves C1, C2, C3, and

C4; each of which does represent a

function.

C1,

C1

C2,

C2

C3,C4;

C3

C4

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2 4 6 8 10

2

16

4

6

8

10

12

0 14

14

16

18

12 18 20 22 24

20


Notice that the line x = 5 intersects each of the curves, C1, C2,

C3, and C4 at one and only one point (e.g., it intersects C4 only at

the point S, etc.)

C1

C2

C3

C4

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x = 5

S

P

Q

R


Not Every Rule is a FunctionNot Every Rule is a Functionnext

As an application of what we have just discussed, consider the following…

What is incorrect about the What is incorrect about the following statement?following statement?

“Let f be the function that assigns to every (non-negative)

number its square root.”

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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 32 = 9, it is

also true that (-3)2 = 9.

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That is, a number such as 9 has two square roots; namely, 3 and -3.


To avoid this ambiguity we can represent f as the union of the two functions g and h, where…

Note

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g(x) = + ; image of g = {y:y ≥ 0}x

In other words, g assigns to any non-negative number its

non-negative square root.

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For example… g(9) = 3

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h(y) = - ; image of h = {y:y ≤ 0}x


And on the other hand…

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h(x) = - ; image of h = {y:y ≤ 0}x

means that h assigns to any non-negative number its

negative square root.

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For example… h(9) = -3


So in terms of g and h, f(x) = g(x) U h(x).

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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.

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For example…

g(x) = + ; image of g = {y:y ≥ 0}x

f(9) =+3 = g(9)-3 = h(9){ = g(9) U h(9).= g(9) U h(9).

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f(x) = g(x) U h(x).

From a graphing point of view…

The curve C1 is the graph of g,

x

y

and the curve C2 is the graph of h.

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C1

C2

(9,3)

(9,-3)


Graphing the Union …

C, which represents the graph of f, is the union of C1 and C2.

x

ynextnextnext

(9,-3)

(9,3)


Graphing the Intersection…

x

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x = 9


P

The vertical line x = 9 intersects C at the two points P(9,3) and Q(9,-3).

However, P is only on C1

(9,3)

(9,-3)Q

C1 (y = + )x

and Q is only on C2.

C2 (y = - )x

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To avoid confusion we agree to let Notenext

For example, when we write

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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 4.±±

In this context, when we say the square root of x, we mean the principal square root of x.

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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

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Historical NoteHistorical 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”.

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Keystone Problems for Keystone Problems for Lesson 17 Part 3Lesson 17 Part 3

The function f, mentioned in Problem 1b, is defined by the rule f(x) = 3x + 1

Problem #1bProblem #1b

Suppose the line L is the analytic continuation of the geometric graph of f.

Is the point (20,61) on the line L?

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Solution for Problem #1bSolution for Problem #1b

A point (x,y) is on this line L if and only if…

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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.


Algebraic & Geometric

Relationships

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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.


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.

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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.

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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.

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Knowing that the equation of the line is y = 3x + 1 tells us much

more than that (20,61) is on the line.


More generally, all points (20, y) for which y < 61 lie below the line.

Note

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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.

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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.

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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…

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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


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Note

y = 3x + 1

y < 3x + 1

y > 3x + 1

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Pictorially…

Finally, the same discussion applies to any line when written in the form y = mx + b.

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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.


Namely…

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

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