Functions Functions play an important role in mathematics as well as computer science. A function is a special type of relation. So what s a relation? A relation, R, from set A to set B is defined as arb where a A, b B and we say a is related to b. Howdowewritea is not related to b? Eample I = {(a, b) a {, 2, 3, 4, 5},b is a letter in the name of the number a. } 2Rw Ro Eample 2 How many possible (a, b) pairs are there? How many are there really? There (a, b) pairs are elements in a set A B. They are ordered pairs. Thus (a, b) =(c, d) iff what? We define the product of sets as A B = {(a, b) a A, b B}
Eample 3 If A = {, y, z},b = {2, 3} find the set A B. How many elements are there? Can we make that a rule if we call A = the number of elements in set A? Eample 4 What s R 2? Eample 5 What s [0, ] [, 3]? Eample 6 Sketch the above eamples. Do the coordinate diagram last. Now we can say arb A B Why? Define Range and Domain of a relation and illustrate them on an arrow diagram and a graph for the relation arb = {(, 2), (, 3), (, ), (w, 3), (m, n)} Why is that not a good computer thing we drew? That s why it s not a function, it s just a relation. We mentions equivalence relation last quarter but they are important here again so I will repeat the definition.
. Equivalence Relations A relation arb A A that satisfies these properties Refleive ara Symmetric If arb then bra Transitive If arb and brc then arc is called an equivalence relation. Eample 7 a b mod 3 is an equivalence relation, justify. Eample 8 Is less than, <, an equivalence relation? What about? Eample 9 Find an equivalence relation for the set {, 2, 3, 4, 5, 6, 7, 8, 9, 0} In each equivalence relation that you found for the last eample, you partitioned the set into equivalence classes If arb is an equivalence class on A, doesit always partition A does it always partition the set A into equivalence classes A i? Note that A i A j = A = n i=a i
How can we name these equivalence classes? [a] ={ A, ar} So we can call [a] the set of elements equivalent to a. Eample 0 Describe the set of equivalence classes of the relation a b mod 4 The set of all equivalent classes [a] is called the quotient set of A by R and is denoted by A/R Eample Describe the above set of equivalence classes as a quotient set. Eample 2 If A = {a, b, c, d, e, f, g} and the equivalence relation {(a, b), (b, c), (e, j), (e, g), (d, d)} arb what s A/R? Eample 3 If A = Z + and arb = {a and b share a factor} what s Z + /R?
.2 The Partition Theorem Let R be an equivalence relation on a set A. The quotient set of A by R is a partition of A. Every element of A belongs to an equivalence class (only one) and if [a] 6= [b], then[a] and [b] are disjoint. Prove the partition theorem..2. Homework 5. a,b,c, 2b,d 3a,b,c, 4, 6a,b,c, Submit 4, 6a for Monday Read section 5.
So what makes a relation a function? If we think of a function as a machine that requires an input and then produces an output is the eample arb = {(a, ), (b, ), (e, 3), (e, 2), (c, 4)} auseful machine? That relation above is not a function because we don t have one answer to a question, that is, there is two outputs for one input. So a function maps every element in the domain onto only one element in the range. The range is a subset of the codomain. What s the difference between the codomain and the range? Eample 4 Let the function in question be f : Z Z + be defined as f() = 2. What is the domain, range and codomain in this case? What is the image of 3? What is the value of f at =2?We can say that f maps 2 into what? What elements are in the codomain but not in the range? Read the Notes 5. and 5.2 in the tet..3 Split Functions Sometimes called piecewise functions.
Eample 5 f() = < 0 + 0 <2 2 2 < Define the domain, range, and codomain of the above split function. Eample 6 What is the image of, 0,, 5 under the function shown above. Of course many functions do not map numbers to numbers. There are instances when the domain is the set of character strings. Eample 7 Call the function CNT assign to a character string the number of words in the string. That is CNT( You are the weakest link, goodbye. ) = 6. What are the domain and range? Of course a function doesn t have to have just one input. Eample 8 f(, y) = 2 sin y
20-4 4-2 2 y 2-20 -2 4-4 What are the domain, range, codomain of the function? Find f(2, π)? Eample 9 What about MAX(, y) which is defined as the larger of the two numbers? MAX : R R R Eample 20 Call DED : S Z + {yes, no} the function that determines whether the integer is larger than the length of the character string. Find DED( Usonian Houses,6).3. Homework Section 5.2 read all, #-, submit 6, 8a for Tuesday.
2 Graphs of Functions Graphs of functions are visual representation. We have already seen coordinate diagrams, used when the domain and range of the function are finite sets. We have seen arrow diagrams. The graphs you are probably most familiar with are of functions where the domain and range are subsets of the real numbers. Eample 2 Graph the function y = 2 7 5 4 3 2-0 2 3 - What are the domain and range? The function y = 2 is an eample of a polynomial. The general polynomial can be written as y = a 0 + a + a 2 2 + a 3 3 +...a n n nx = a i i i=0 and is considered an nth order polynomial. We can see that the graph above has two roots, that is, s that make the y value zero. This property can be
etrapolated to the nth order polynomial. That is, an nth order polynomial has at most n roots. It might have less. Also, for any given nth order polynomial, if n is odd, the polynomial has at least one root and its range is all the reals. The domain of all polynomials is all the real numbers. Eample 22 Sketch an eample of an 5th order polynomial and a first order polynomial. Eample 23 Give an eample of a polynomial with no roots. 2. Vertical line test We can determine whether a graph represents a function by using the vertical line test. If any vertical line crosses the graph only once, then the graph is a function. Why can I say the graph is a function? Read eample 5.6 of the tet and remember the open and closed dots we first learned with split functions. 2.2 Intercepts and Asymptotes A graph can intercept the or y ais. A graph can have many intercepts, roots, but at most one y
intercept. To find the intercept, set y=0, to find the y intercept, set =0. Eample 24 Find all the intercepts of the function y = 2 +7 +2and use them to sketch the graph of the parabola. An asymptote of a graph is a line that the graph approaches as the magnitude of grows ever larger, ( horizontal asymptote) or an value that when approached makes the y value approach infinity, (vertical asymptote). Eample 25 Find the asymptotes of the function y = 2 2 4 2.3 Increasing and Decreasing Increasing and Decreasing are always read left to right and can refer to an interval. y = 2 is increasing on the interval (0, ) anddecreasingontheinterval (, 0). What happens at =0? Definition of increasing on an interval, I. Definition of decreasing on an interval, I.
Eample: Is the absolute value function increasingontheinterval[ 2, 3]? Note: website. Thereisagraphingfunctiononthe Consider the function y = sin( ) 0.8 0.6 0.4 0.2 - -0.8-0.6-0.4 0-0.2-0.2 0.2 0.4 0.6 0.8-0.4-0.6-0.8 -
3 Even and Odd functions These terms refer to the symmetry of a function. For eample y = 4 is an even function because it is symmetrical about the y-ais. Whereas y = 5 is odd because you can fold it on the -ais and then the y-ais in a kind a odd way. ( or 80 degree rotation) The formal math definitions are: Even Function Odd Function A function could also be neither even nor odd such as the line y =2 +3 Eample: Prove the function < 2 f() = 6 2 2 2 < is even. 3.0. Homework Section 5.3, #2-6, Submit 4,6 for Thursday before the quiz.. 4 Classes of Functions We have already introduced the class of functions called polynomials. Another is
4. Rational Functions Rational functions are made from the ratio of two polynomials, P (),Q() f() = P () Q() Remember that we might have vertical asymptotes when Q() =0. Eample 26 Find the vertical and horizontal asymptotes of the rational function y = 3 +3 2 3 +3 2 + 4.2 Limits To introduce the idea of limits, consider the function f() = What happens to the y-value of y = f() as the -value gets closer and closer to? We can t just get the y-value for =because f() is not defined. We can tdividebyzero. Butwecanputinan-value close to, say =.00. We could get closer than that really, try =.000. In fact any number close to that you choose, I can get one closer. What we want to know is what does the y-value look like it is becoming as the -value moves closer and closer to
. If we chart it we get f(.) = f(.0) = 0 f(.00) = 00 f(.000) = 000. f(.00000000) = 00000000 So it looks like as the -value gets closer to, the y-valuegoestoinfinity. In the chart above we are only approaching =from one side, from the positive numbers. If we approach from the negative side say f(0.99) andsoonweseethatitlookslikethey-value approaches infinity. Try it yourself with a calculator. The notation of limits looks like this: lim a f() =L where a is the number that the -value is getting closer and closer to. f() is the function in question and L is the number that the y-value looks like it is approaching. Note that the -value never really has to get to a and the y-value never really has to get to L. L canbeafinite number or it can be undefined or ±. We say the limit of f of as approaches a for lim a f(). Using this notation with our f() we can
write lim =+ Look at eamples 5.23 and 5.24 in the tet and read section 5.4 entirely. If we have the graph of a function we can just check what the y-value of the function looks like it is approaching. We can see from the graph that as the -value approaches from both sides that the y-value goes to +. What about the limit lim Now as approaches infinity what does the y-value approach? If you try the chart method it seems that the y-value approaches. But the chart method is
not a proof, it s just an indication of the answer. To solve the limit computationally multiply the function by = and rewrite the problem as lim = lim Now we see that as the -value get larger and larger the part goes to zero. (Try it with a calculator) Therefore lim = 0 = so the limit approaches. This is confirmed by the graph above. Eample Q? Solve lim +2 2 Eample Q? Solve 2 2 lim 3 + 2 A. If we plug = into the numerator we
get. If we plug = into the denominator we get. Thereforef(a) =, a Problem Limit. Because it is a polynomial over a polynomial and,one technique is to multiply top and bottom by 3 3 which is chosen because the highest power of in the whole function is 3. We also see that we are really just multiplying by, so we re not changing the value of the function, just it s appearance. Our result will be more algebraically convenient. 2 2 lim 3 3 + 2 = lim 2 3 + 3 2 2 3 The beauty of this technique is that now all the s are the denominators of fractions and we see from the list that = 2 = = 2 =0. Therefore lim 2 0 0 3 + 2 = lim +0 0 = lim 0 =0 2 3 Wecanuseacalculatortocheckiftheansweris correct. Of course we can t put into our calculator but we could try = 000 and see if we get a result close to zero. Try it and see. Eample
Q? Solve µ 2 lim ( 3) 2 A. From the list we see that =0and we know that 3=, so we have a Problem Limit in the form of (0) µ ( ). Note that 2 2 6 lim ( 3) = lim = lim 2 6 =2 0=2 Some important algebraic techniques to handle limits taught thru eample: Eample 27 lim 0 +4 2 Eample 28 lim 2 ( ) 2 2 In the end we will use limits to evaluate Horizontal asymptotes for rational functions. The rational function f() = P () Q() has the horizon-
tal asymptote y = b if lim ± 4.3 Eponential Functions P () Q() = b We are talking about y = a functions where a>0. We have three cases, 0 <a< a = a> Graph them What does it mean when we raise 2? We know what we mean when =3, that is raised to a positive integer. We know what we mean when we raise a number to a negative integer. 2 5 What about rational numbers? 2 2 5 that is in general a p q = q p p a = q (ap ) and if it s a negative rational number, we invoke
the reciprocal idea. Is every type of eponent covered? No, we left out irrational numbers as eponents. What to do? If we graphed what we have now for say, y =2 we get 30 25 20 5 0 5 0-4 -2 2 4 4.4 Laws of Eponents a +y a y (a ) y (ab) 5 π 0 s country cousin e Everybody knows π as an irrational number. It has a movie, books about it. But e =2.78.. is another special irrational number that comes into its own
when we talk about eponential functions. That is, y = e function occurs in mathematics in every subdiscipline. If we graph the curves, y =2.5,y = e,y =3, 4 3 2 0-2 -.5 - -0.5 0.5.5 5. Two definitions of the number e As h approaches zero e h = h As n approaches infinity µ + n n = e What s so special about the curve y = e? It is the only curve (ecept) that is it s own slope function, the tangent slope of the curve is the y-value. 5.. Homework Section 5.4, - 7, Submit 4 and 6.
6 New Functions From Old Functions We can make new functions from old ones by adding them, subtracting them, multiplying them or dividing them. (f + g)() =f()+g() (f g)() =f() g() (fg)() =f()g() ( f )() =f() g g() Eample Q? If f() = 2,g() = +2find ( f g )(). A. ( f 2 g )() = +2, note that both original functions, f(),g(), can accept any -value ( Remember Domain) but the new function can not accept = 2 because it would mean dividing by zero. Another way to combine old functions and make new ones is through function composition. That is, putting one function inside another. When I see h() =sin( +2)I see the function g() = +2 inside the function f() =sin(). I can write h() = f(g()) = (f g)(). The g() function is inside the f() function. Eample
Q? Find f(g()) and g(f()) if f() =cos() and g() = 2 +3 A. If f() =cos() then f(a) =cos(a) and f(g) =cos(g). So f(g()) = cos(g()) = cos( 2 +3). Nowusingthesametypeofarguments,g() = 2 +3 so g(a) =A 2 +3A. Therefore g(f) =f 2 +3f and g(f()) = (f()) 2 +3f() =(cos()) 2 +3cos() 7 Inverse Functions We know what it means for a relation to be a function. Every input maps to only one output, it passes the vertical line test. But not every function has an inverse. A function has no inverse if there are outputs that are mapped to by different inputs such as in the case of y = 2. If a function has an inverse then we can determine the input if we know the output. For eample if the function f() = 3 7 gave an output of 20 what was the input? What about if the output was 5.67? How are you solving this? The function you are using is the inverse, f (). Find it algebraically. Graphically, a function has an inverse if it passes the horizontal line test. A function that has an
inverseissaidtobeaone to one function. That means every one input goes to only one output. In math we can write 6= 2 f( ) 6= f( 2 ) Prove the eample above is a one to one function. Begin by assuming that 6= 2 but f( )=f( 2 ) and prove by contradiction. Of course that makes good sense because we already found the inverse and we can draw the graph and show it passes the horizontal line test. So let s say a function, y = f() is one-to-one with domain A and range B. The inverse function, f (y) =, has domain B and range A. Notice how the range and domain switch places. So what s f (f(a)) = f(f (b)) = and where does the a and b come from? So we know how to algebraically get f () from f() but how do we get the graph of the inverse from the original graph?
As an eample of the technique, sketch the graph of y = e and then find its inverse.
8 Logarithms The logarithm function, y =log b () is defined for all b>0,b6=, where b is the base. The graph looks like 0 0.5.5 2 2.5 3 - -2-3 for all b s. We can see that the range is all the reals and the domain is (0, ). This introduces a new complication into our find the domain of the function questions. Before this we worried about not dividing by zero and not taking the square root of a negative number. Wealsohavetomakesurenottotakethe log of zero or a negative number. So what s logarithm? Well, when I write log b () I am saying What do I have to raise b to, to get?. In other words y =log b () is equivalent to b y = Ifthereisnobasewritten,itisassumedtobe base 0, like on your calculator. That is, log(00) = 2.
Eample Q? Evaluate log 2 (6). A. What number do I have to raise 2 to to get 6?, Well 4, since 2 4 =6 Eample Q? Evaluate log 3 ( 27 ). A. What number do I have to raise 3 to to get 27?, Well it must be negative to make the fraction over and note that 3 3 =27, therefore log 3 ( 27 )= 3. That is 3 3 =27. Eample Q? Evaluate log 29 (29). A. What number do I have to raise 29 to to get 29?, Well, since 29 =29. This would have been the answer for any base, not just 29. Eample Q? Evaluate log 7 (). A. What number do I have to raise 7 to to get?, Well 0, since 7 0 =. Thiswouldhavebeen the answer for any base, not just 7.
8. The Natural Logarithm, ln() The logarithmic function y =log e () occurs so often that it was given its own symbol, ln(). That is ln() =log e (). The rules for regular log s apply for ln() also. The function y =ln() is the inverse function to y = e, that is, they undo each other. ln(e )= e ln() = We will use this fact to solve for. Try taking the ln(e 7 ) on your calculator. Your calculator probably only has log base 0, log(), and base e, ln(). Eample Q? Solve for, e 2 2 =7 A. Take the ln() of both sides. e 2 2 =7 ln(e 2 2 )=ln(7) 2 2=ln(7) 2 =ln(7)+2 = ln(7) + 2 2 Note that ln(7) can be approimated on a calculator, it s just a decimal number and is treated as such in the above algebra. Eample
Q? Solve for, ln( 7) = 29 Eample Q? Find the domain of the function y = ln( 2) A. There is no root sign. In order not to divide by zero, ln( 2) 6= 0 e ln( 2) 6= e 0 2 6= 6= 3 Now we just have to worry about only taking the ln() of a strictly positive number, that is, 2 > 0, > 2. Therefore the domain of the function is (2, 3) (3, ). Section 5.5, #,6,7a,0,,2. Submit 6a,0a,2a, d, e, f Friday.