The Game of Algebra or The Other Side of Arithmetic The Game of Algebra or The Other Side of Arithmetic © 2007 Herbert I. Gross by Herbert I. Gross & Richard.

© 2007 Herbert I. Gross The concept of functions, including inverse functions and composition of functions, gives us an effective way to explain the informal process of “undoing” in a more mathematically precise way. next For example, we have viewed subtraction as being “unaddition”. The more formal view is to refer to subtraction as being the inverse of addition. There is a strong connection between the word “inverse” as it is used here and the word “inverse” as it is used in the term “inverse function”.

© 2007 Herbert I. Gross For example, to “undo” adding 3 we subtract 3. That is: if we add 3 to a given number and then subtract 3 the result is the given number. In less formal terms, “we are back to where we started from”. next In the language of functions, the instruction “Add 3” can be represented by the function f where f(x) = x + 3. The instruction “Subtract 3” can be represented by the function g where g(x) = x – 3. So if our input is 7, we see that f(7) = 10, and g(f(7)) = g(10) = 7. In a similar way, we see that g(7) = 4 and, f(g(7)) = f(4) = 7.

© 2007 Herbert I. Gross There was nothing special about our choice of 7 as our input. That is: with f and g defined by f(x) = x + 3, and g(x) by g(x) = x – 3; for any number x, f(g(x)) = g(f(x)) = x. More specifically… f(g(x)) = f (g(x)) and… next (x – 3) = (x – 3) g(f(x)) = g (f(x)) (x + 3) = (x + 3) next + 3 = x – 3 = x next

© 2007 Herbert I. Gross The composition of these two functions, g and f, is itself a function, and it is a function that basically “does nothing”. next That is: if we let I denote fog (or in this case, equivalently gof) we see that I(x) = x. For this reason, we refer to I as being the identity function.

© 2007 Herbert I. Gross Just as multiplying a number by 1 doesn’t change the number, composing a function with I doesn’t change the function. That is, for any function, f … next If f(g(x)) = g(f(x)) = I(x) = x for all x in the domain of f and the domain of g, we call f and g inverses of one another and write f = g -1 and g = f -1. Notes f(x)f(x) f(x)f(x) I(x)I(x) I(x) x I( ) = ( ) and f( ) = f( ) next Definition D

next © 2007 Herbert I. Gross In essence, we obtain the inverse of a function (if there is an inverse function) by interchanging the input and the output (that is, the domain and the image). next For example, suppose that… f(x) = x + 3 If we interchange x and f(x) the equation becomes… x = f(x) + 3 and if we subtract 3 from both sides of our equation we obtain… f(x) = x – 3 next

© 2007 Herbert I. Gross The “tricky part” is that f(x) in the equation f(x) = x – 3 is not the f(x) in the equation f(x) = x + 3; rather it is its inverse because we interchanged the roles of x and f(x). In other words… If f(x) = x + 3, then f -1 (x) = x – 3 y = f(x) = x + 3 …and if we then solve for x in the above equation we see that… x = y – 3 = g(y) = f -1 (y) next The confusion results from the tradition that we usually denote the input of a function by x. To avoid this confusion we may let y = f(x) in the equation f(x) = x + 3 to obtain…

© 2007 Herbert I. Gross Not every function has an inverse. For example, suppose we define a function on the set of all 3-digit numbers such that the output of the function is the sum of the 3 digits. Thus, for example, f(123) = 1 + 2 + 3 = 6 However, there are several other 3-digit numbers the sum of whose digits is 6. They are… 105, 114, 123, 132, 141, 150, 204, 213, 222, 231, 240, 303, 312, 321, 330, 402, 411, 420, 501, 510 and 600. So for example, 123 ≠ 510, yet f(123) = f(510) = 6. next

© 2007 Herbert I. Gross However, if the function had had an inverse, it would mean that for each output there would have been only one input that yielded that output. In more mathematical terms, it means that if a ≠ b, f(a) ≠ f(b). Thus, in this case f -1 doesn’t exist. next If m ≠ 0, the linear function f, defined by f(x) = mx + b always has an inverse. Namely, if m ≠ 0, its graph is either always rising or always falling. That is, if m > 0 the line is always rising, and if m < 0 the line is always falling. In either case, this means that no two inputs can have the same output.

© 2007 Herbert I. Gross next To obtain the inverse of an “invertible” function, we interchange the input and the output. In terms of the graphs, we interchange the points (x,y) and (y,x). In general, these two points will be different. (5,3) (3,5) x y For example, the point (3,5) is not the same as the point (5,3) next

© 2007 Herbert I. Gross In fact, the only time that these two points will actually be the same is when x = y. Hence, the points of intersection of the graphs of y = f(x) and y = f -1 (x) will always be on the line whose equation is y = x. More specifically, the two graphs will be symmetric with respect to the line y = x. next In other words, if we look at the graph y = f(x) through a mirror that is placed on the line y = x, the mirror image of this graph will be the graph y = f -1 (x).

(0, - 3) ( - 3,0) (0,3) (3,6) (6,9) (3,0) (6,3) (9,6) y = x + 3 y = x y = x – 3 This is illustrated below with the graphs y = x + 3, y = x – 3,and y = x. next

For further practice, let’s draw the line… …to obtain the line, y = f -1 (x), …and reflect it about the line, y = x, y = f(x) = 2x + 3, ( - 3, - 3) (0,3) (3,9) (3,0) (9,3) y = 2x + 3 y = f -1 (x) y = x next Note that… x – 3 2 f -1 (x) = next

© 2007 Herbert I. Gross If m is negative, a similar analysis holds. Namely if m is negative it means that as x increases, f(x) decreases. Hence again, no two different values for x can yield the same value for f(x). next Note Geometrically, it means that any horizontal line intersects the straight line that represents the graph of f at one and only one point.

next © 2007 Herbert I. Gross If a function is continuous (that is, if its graph is “unbroken”) then it has an inverse if and only if its graph is either “always rising” or “always falling”. Generalization If its graph is “always rising” we say that the function is monotonically increasing; and if its graph is “always falling” we say that the function is monotonically decreasing.

next © 2007 Herbert I. Gross In the language of functions we say that f is monotonically increasing, if whenever a > b, then f(a) > f(b) In other words: as x increases f(x) also increases... Key Point And we say that f is monotonically decreasing if whenever a > b, then f(a) < f(b). That is, as x increases f(x) decreases. next

© 2007 Herbert I. Gross For example, with respect to the function f where f(x) = 2 x, notice that as x increases so also does f(x). In other words, the curve y = 2 x which represents the graph of f is always rising. Hence, no horizontal line can intersect the graph at more than one point.

next © 2007 Herbert I. Gross In geometric terms, our above discussion essentially means that if the curve that represents the continuous function is sometimes rising and sometimes falling, then the function will not have an inverse. Key Point Let’s illustrate this in the case for which f is defined by f(x) = x 2. Its graph is the curve y = x 2. Notice that while every vertical line intersects this curve at one and only one point, every horizontal line that is above the x-axis intersects the curve in two places. next

© 2007 Herbert I. Gross Since f(x) = x 2 and since there are two values of x for which x 2 = 4, it means that f -1 doesn’t exist. Key Point next In other words, if this function had an inverse, then for each output y there would be one and only one value of x for which (x,y) belongs to the graph.

next © 2007 Herbert I. Gross Notice that every vertical line intersects this curve at one and only one point. next ( - 2,4)(2,4) x y This is illustrated for the line x = 2 which intersects the curve only at the point (2,4), and the line x = - 2 which intersects the curve only at the point ( - 2,4) x = 2 x = - 2 y = x 2

next © 2007 Herbert I. Gross next ( - 2,4)(2,4) x y On the other hand, if instead, we start with the horizontal line y = 4, y = 4 we see that it intersects the curve at two different points, namely (2,4) and ( - 2,4). y = x 2

next © 2007 Herbert I. Gross At first glance it may seem that not all vertical lines will intersect the curve y = x 2. next Notice, however, that no matter how large the number c is, the line y = c will intersect the curve y = x 2 at the point (c,c 2 ) Aside

next © 2007 Herbert I. Gross Although the line y = 4 intersects the curve in two points, notice that we can represent y = x 2 as the union of the two curves y = x 2 where x is non-negative, next ( - 3,9) ( - 2,4) (3,9) (2,4) (0,0) x y next and y = x 2 where x is negative. C1C1 C2C2

© 2007 Herbert I. Gross The curves C 1 and C 2 are the graphs of the functions g and h where g(x) = x 2 when x is non-negative and h(x) = x 2 when x is negative. In other words, if g and h are defined as above, g is a monotonically increasing function and h is a monotonically decreasing function. next Therefore, the inverses of both g and h exist. next

© 2007 Herbert I. Gross In summary… next y = h(x)y = g(x) y ( - 3,9) ( - 2,4) (3,9) (2,4) (0,0)(0,0) x C1C1 C2C2 The line y = 4 intersects the curve C 1 only at the point (2,4), and intersects the curve C 2 only at the point ( - 2,4). y = 4

next © 2007 Herbert I. Gross Independently of whether the graph of a function is a curve that is always rising or always falling we may still interchange its x- and y-coordinates. Whenever we do this, the resulting curve will be the mirror image of original curve with respect to the line y = x, but it might not represent a function. To make this idea more concrete let’s look specifically at the curve x = y 2 which represents the mirror image of the curve y = x 2 with respect to the line y = x. In terms of functions, this curve represents the function x = [f(x)] 2. next

© 2007 Herbert I. Gross In our “plain English” version when we define the function f by the equation x = [f(x)] 2 we are defining the function implicitly. Note That is: we are not explicitly telling what the program is, but what we are saying is that the program has the property that for each input the square of the output is always equal to the input. next

© 2007 Herbert I. Gross The concept of implicit and explicit is not restricted to the study of functions. Note For example, when we say that a number is 4 less than 7, we have implicitly defined the number 3. next

© 2007 Herbert I. Gross Suppose we want to draw the graph that represents the equation… The Curve x = y 2 x = j(y) = y 2 Recalling that y 2 cannot be negative, we see from the above equation that for any value of y, x must be at least as great as 0. Hence, the graph exists only to the right of the y-axis. next

© 2007 Herbert I. Gross To locate points on the curve x = y 2, it’s probably less cumbersome to pick values for y and then to solve for the corresponding value(s) of x. The Curve x = y 2 For example… y y2y2y2y2x(x,y)000(0,0)111(1,1) -111(1, - 1)244(4,2) -2-244(4, - 2)399(9,3) -3-399(9, - 3)416 (16,4) -4-416 (16, - 4) next

next © 2007 Herbert I. Gross The fact that for each (positive) value of x there are two corresponding values of y means that x = y 2 is not the graph of a function. The Curve x = y 2 y = ± √ x More explicitly, if we take the square root of both sides of the equation x = y 2, we obtain the equivalent equation… next

© 2007 Herbert I. Gross next x y (0,0) (4,2) (9,3) (4, - 2) (9, - 3) x = y 2 While the line x = 4 intersects the curve x = y 2 at both (4,2) and (4, - 2); “y = + √ x only at (4, - 2). y = - √ x ” it intersects only at (4,2) and y = + √ x y = - √ x x = 4 next

© 2007 Herbert I. Gross More generally, we see that the curve D 1 defined by y = + √ x is always rising. Hence, it represents a function which we will denote by k p. That is: k p (x) = + √ x In a similar way, we see that the curve D 2 defined by y = - √ x is always falling. Hence, it represents a function which we will denote by k n. That is: k n (x) = - √ x next

More specifically… g(x) = x 2 when x is non-negative and h(x) = x 2 when x is negative In Summary… next What we have shown thus far is that the function f, defined by f(x) = x 2 does not possess an inverse. However, it is the union of two functions, g and h, both of which have an inverse. © 2007 Herbert I. Gross

An Historical Note Given a continuous curve drawn at random; it is very unlikely that it is either always rising or always falling. Therefore, if a continuous curve is drawn at random, it will probably not represent a function that possesses an inverse. For this reason, until relatively recently, it was common to talk about multi-valued functions. In that context if we let f(x) = x 2, we would have said that f -1 existed as a multi-valued function; and the two functions k p (x) and k n (x) would have been referred to as single valued branches of f -1. © 2007 Herbert I. Gross next

A Historical Note Nowadays, however, we use the word “function” to mean “single valued function”; and we use the term “relation” rather than “multi-valued function”. That is, we would refer to f(x) = ± √x The distinction between a relation and a multi- valued function is not too important because every relation (that is, a multi-valued function) can be represented by the union of two or more functions, each of which has an inverse. © 2007 Herbert I. Gross next This was illustrated in our above discussion when we talked about trying to find the inverse of f in the particular case where f(x) = x 2. as being a relation.

An Enrichment Discussion Prelude Suppose we had “invented” addition but had not yet “invented” subtraction. In this case, we would know that the function f, defined by f(x) = x + 3 must have an inverse. Namely because no two different values of x have the same image, we see that once we know the value of x + 3, we can uniquely determine the corresponding value of x. In terms of a graph, we could construct f -1 by first drawing the line y = x + 3 and then reflecting it about the line y = x. The resulting line would be the graph of f -1. © 2007 Herbert I. Gross next

An Enrichment Discussion Notice that since we can draw its graph, the function f -1 exists even if we don’t bother to give the function a specific name. Thus, to solve an equation such as 3 + x = 12 using a calculator, we would translate this indirect computation into a direct computation by rewriting it in the equivalent form 12 – 3 = x. © 2007 Herbert I. Gross next However, as we all know, it turned out that we “invented” the minus sign and defined f -1 by f -1 (x) = x – 3.

An Enrichment Discussion Of course this discussion might seem boring to us because we’ve know for a long time, even if not in these exact words, that subtraction is the inverse of addition. © 2007 Herbert I. Gross next

An Enrichment Discussion The above discussion applies almost verbatim to any function f, provided that f is either monotonically increasing or monotonically decreasing. In particular it applies to the monotonically increasing function f where f(x) = 2 x. However, we are most likely not nearly as comfortable discussing exponential growth and its inverse as we are when we are discussing addition and its inverse. So let’s construct f -1, when f(x) = 2 x. © 2007 Herbert I. Gross next

An Enrichment Discussion By way of review, starting with f(x) = 2 x we can choose various values for x and compute the ordered pairs (x,2 x ). In this way we see that the points ( - 1, 1 / 2 ), (0,1), (1,2), (2,4), (3,8)... belong to the curve y = 2 x (which represents the graph of f). By interchanging x and y we see that the points ( 1 / 2, - 1), (1,0), (2,1), (4,2), (8,3)…belong to the curve that represents the graph of f -1. Or looking only at the geometric graph, we start with the curve y = 2 x and reflect it, point by point, about the line y = x. In this way the curve that is thus generated represents the graph of f -1 © 2007 Herbert I. Gross next

Pictorially… (1,2) (0,1) (1,0) (1, -1 / 2 ) (- 1,1 / 2 ) y = x (3,8) (2,1) (8,3) next Thus, f -1 exists whether or not we give it an explicit name. It turns out that the name we give to f -1 in this case is f -1 (x) = log 2 x f(x) = 2 x f -1 (x) = log 2 x © 2007 Herbert I. Gross (2,4) (4,2) next

An Enrichment Discussion There was nothing special about choosing the base to be 2 and then showing that if f(x) = 2 x, then f -1 (x) = log 2 x. © 2007 Herbert I. Gross next if f(x) = 2 x, then f -1 ( x ) = log 2 x b b Because of place value notation, we often choose b to be 10. In this case we omit the subscript and simply write log x rather than log 10 x. Because of place value notation, we often choose b to be 10. In this case we omit the subscript and simply write log x rather than log 10 x. Thus, if we replace 2 by b, we obtain the more general result… next

An Enrichment Discussion For example… 2 3 =8 means the same thing as log 2 8 = 3 5 2 = 25 means the same thing as log 5 25 = 2 4 -2 = 1/16 means the same thing as log 4 1/16 = - 2 10 3 = 1,000 means the same thing as log 1,000 = 3 © 2007 Herbert I. Gross next Or with a shift in emphasis… log 6 36 = 2 means the same thing as 6 2 = 36 log 3 81 = 4 means the same thing as 3 4 = 81 log 2 1/8 = - 3 means the same thing as 2 -3 = 1/8 log 10,000 = 4 means the same thing as 10 4 =10,000

An Enrichment Discussion If we try to use the calculator to find the value of x if, say, x = log 3 15, we discover that there is no log 3 key. It would have been nice if there were such a key because we could then enter 15 and press the log 3 key to obtain the value of x. © 2007 Herbert I. Gross next However, as we discussed in our previous lesson on exponential functions (Lesson 19), most calculators have a “log x” key. Reminder… The relationship between 10 x and log x is that if f(x) = 10 x, f -1 (x) = log x. next

Solving the Equation b x = c Let’s use a calculator that has a “log x” key to solve the equation 10 x = 7, © 2007 Herbert I. Gross next Where b and c Are Positive Constants In this way, we obtain the result that to 4 decimal place accuracy, x = 0.8451. In other words, 10 0.8451 = 7. 0.8451 log x 7 we enter 7 on the calculator and then press the “log x” key.

The major problem is… © 2007 Herbert I. Gross next 7 x = 28 by the equivalent equation… and by our rules for exponents, we can replace the above equation by the equivalent equation… 10 0.8451x = 28 next (10 0.8451 ) For example, suppose we want to find the value of x for which 7 x = 28. Knowing that 7 = 10 0.8451, we may replace the equation… What happens when the base is not 10? What happens when the base is not 10?

© 2007 Herbert I. Gross next Using our calculator we see that… log 28 = log 28 = 1.447158 next Solving the Equation b x = c 0.8451x 10 0.8451x = 28 0.8451x Since log 28 means the power to which 10 has to be raised to obtain 28 and since the above equation tells us this power is 0.8451x, we see that the above equation may be rewritten as… 2 8 1.447158 log x

So replacing log 28 by 1.447158 in the equation… © 2007 Herbert I. Gross next log 28 = 0.8451x we obtain the equivalent equation… 1.447158 = 0.8451x next Solving the Equation b x = c And from the equation above, we see that… x = 1.447158 ÷ 0.8451 = 1.71259

As a check we see that 7 1.71259 = 28.00957... © 2007 Herbert I. Gross next x = 1.447158 ÷ 0.8451 next Checking the Solution for the Equation b x = c can be rewritten as… If 7 x = 28, x = 1.447158 ÷ 0.8451 log 28 log 7 Since 1.447158 = log 28 and 0.8451 = log 7, the derivation that took us from the equation 7 x = 28 to the equation…

For further practice, let’s apply the above formula with b = 8 and y = 40 to find the value of x for which… 8 x = 40 © 2007 Herbert I. Gross next Solving the Equation b x = c If 7 x = 28, x = log 28 ÷ log 7 bby, y and 28 by y… If we replace 7 by b… we obtain the more general result…

Using the “log x” key we see that… log 40 = 1.60206 and log 8 = 0.903090 © 2007 Herbert I. Gross next In this case, the formula x = log y ÷ log b tells us that… x = log 40 ÷ log 8 Solving the Equation b x = c To Solve the Equation 8 x = 40 Therefore… X = 1.60206 ÷ 0.903090 = 1.7740 As a check we see that 8 1.7740 = 40.002 next

The Game of Algebra or The Other Side of Arithmetic The Game of Algebra or The Other Side of Arithmetic © 2007 Herbert I. Gross by Herbert I. Gross & Richard.

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The Game of Algebra or The Other Side of Arithmetic The Game of Algebra or The Other Side of Arithmetic © 2007 Herbert I. Gross by Herbert I. Gross & Richard.

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Presentation on theme: "The Game of Algebra or The Other Side of Arithmetic The Game of Algebra or The Other Side of Arithmetic © 2007 Herbert I. Gross by Herbert I. Gross & Richard."— Presentation transcript:

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