20 Calculus and Structures

Transcription

1 0 Calculus and Structures

2 CHAPTER FUNCTIONS Calculus and Structures Copright

3 LESSON FUNCTIONS. FUNCTIONS A function f is a relationship between an input and an output and a set of instructions as to how to obtain the output from a given input. For an input, there must be a unique output (onl one.) If we call the input and the output then we use the notation: = f(). Think of f as a machine that generates a unique value of from the range of the function (possible outputs) for an from the domain (possible inputs). Functions are ver important in mathematics because the form the language in which math is encountered and conveed to others. Just as architects use floorplans, or in recent ears cad programs, the mathematician relates to others through functions. Eample : = f() =. The rule is: The output number is the square of the input number. f() = 0 f(0) = 0 f() = 4 f f() = a f (a) = a f() = () Domain: all real numbers Range: all reals, 0 Eample : = f() = input number. The rule is : the output number is the positive square root of the 0 0 = f(0) = 0 = f() = 4 f = f(4) = a+h f(a + h) = a h Domain: 0 Range: 0 Eample 3: z = f(,) = + -. The rule is : The output number is subtracted from the sum of the squares of the input numbers ( (,) Domain : all pairs (,) (,0) 0 Range : z - (0,) f 0, ) 0 Remark : For the function of Eample 3, an element from the domain is an ordered pair of numbers with the first number named and the second, and the output is a single number labeled z. Remark : Notice that we don t have to place eplicit numbers into a function. However, whatever is the input to the function we must appl the rule to get the output. Also it does not matter what letters we use for the input and output of the function. You can see that s = f(t) or u = f(v) gives the same outputs for given inputs. Calculus and Structures

4 SECTION. Another wa to represent a function is with a table in which several ke elements of the domain are placed on the top row and their corresponding elements from the range are in the row beneath it. For eample, f() = is shown below. Table = f () First of all notice that the same number appears twice in the second row whereas a number onl appears once in the top row. This is because each element of the domain can have onl one value in the range whereas several elements of the domain can map to the same element of the range. Such a function is called a man-to-one function. From this table ou can sketch the curve as shown below, = Notice that since each element in the domain maps to onl one element in the range an vertical line intersects the curve onl once. This is the vertical line test for functions. However, a horizontal line drawn on this graph intersects it more than once which is wh the function is called man-to-one. Net consider the linear function defined b the table, Table f() Can ou guess the rule of this function? Notice that there are no repeating elements in row or row, therefore this is called a one-to-one function. Because there are no repeats in the second row (there are never repeats in the first row) we can define a new function, g() in which the rows are interchanged. Calculus and Structures 3

5 LESSON FUNCTIONS Table g() Notice that f() = 5 while g(5) = and f(0) = while g() = 0. In fact we find that if f() = then g() = for all values of. An function g that has this propert is called the inverse function of f and written b the smbol f -. So I will rename g = f -. We will discuss inverse functions further in Section.7. The graphs of f and f - are drawn together in the figure below, f () = - f () Notice that = f - () is the reflection of = f() in a mirror placed on the diagonal =. This will alwas be true of a function and its inverse. You will notice that the function f() = does not have an inverse because if ou reverse the rows then g ( 4) and g ( ) which cannot happen with a function which can onl have a single value from the range for each element from the domain. So f() = does not have an inverse. However, if we limit the domain of f to onl positive values then it does have an inverse, Table f - () 0 3 And the graphs are shown below. In fact we shall show in Section.7 that f - () =. f () = = f () = Calculus and Structures

6 SECTION. Net consider the function, f() =, Table log log f() = /8 ¼ ½ log Notice that neither row has an repeating values so the rows can be reversed and f has an inverse function, f -. In fact the inverse function is better known as f ( ) log. And if ou draw the functions = and log on the same graph ou will notice again how the reflect in a mirror at =. 8 f () = = f () = log Also notice that if ou add two values from the domain such as and 3 to get 5, the corresponding range values 4 and 8 multipl to give 3. In other words, 5 log 3 log 4 8 log 4 log 8 3 Before hand calculators were invented, this propert of logarithms enabled integers to be multiplied b adding their logarithms. Since addition is less time consuming that multiplication, this shortened man computations. In fact, logarithms have the following properties which ou can verif b choosing eamples from the above table. The corresponding properties of eponentials are listed beside them. Note that when the base of the logarithm is not stated, the logarithm propert holds for an base.. log 0 a 0. logb b a a b c 3. log( a b) loga logb a a 4. b log log a log b a 5. log a b blog a 6. log( a b) log a log b a b c b a b c a c a b c bc c ( a ) a ( a ) ( a b) c a c b b c Calculus and Structures 5

7 LESSON FUNCTIONS 7. b log log b b b 8. log0 a log e a logb a log b log b 0 e Remark 3: If base b is the irrational number e =.78, then log e a is generall written as ln a where ln is the abbreviation for natural logarithm. We postpone a discussion of wh base e is considered to be natural for later. Remark 4: Table 5 enables ou to find log a where a is an number that is an integral power of. But what if ou wanted to compute log 5? You cannot do this with our calculator since it is onl able to compute logs to the base 0 and base e. However Propert 8 of logarithms enables ou compute logs to the base in terms of logs to the base 0 or e. For eample, log0 5 ln 5 log 5.3 log ln 0 Problem : For the following two functions, draw the graphs of the function and its inverse on the same graph. a) f() b) /4 /3 ½ 3 4 f() 4 3 ½ /3 ¼ Consider Epression. This epression is ver important to the stud of calculus and will be discussed in Lesson 4. f ( a h) f ( a) h () Problem : For the following functions compute Epression and simplif it b algebra: a) f() = ; b) f() = ; c) f() = 3 ; d) f() = Problem 3: If f() = +, find and simplif, a) f(t+) ; b) f(t + ) ; c) f(t) ; d) (f(t)) + ; e) f() 6 Calculus and Structures

8 SECTION.. GRAPHS AND FUNCTIONS An curve in the plane can be represented b an equation of the form : f(,) = 0 Eample 4a: f(,) = = 0 or =, a parabola cupped up. Notice that the parabola is characterized b its verte, v. This curve passes the vertical line test in that an vertical line cuts the curve onl once. Therefore, we sa that this is the graph of a function, i.e., = f() where f()= Eample 4b : f(,) = + =0 or = -, a parabola cupped down. = = - v v Fig. 4a Fig. 4b Eample 5: f(,) = 3 = 0 or = 3 is a cubic. This curve is characterized b its inflection point I, the place on the curve where its curvature changes from concave down to concave up. = 3 I It passes the vertical line test so we can sa : = f() where f() = 3 Calculus and Structures 7

9 LESSON FUNCTIONS Eample 6: f(,) = + - = 0 or + = is a unit circle (radius = ). The circle is characterized b its center, c. + = c Notice this curve does not pass the vertical line test (wh?) so it cannot be epressed as the graph of a function. Eample 7: f(,) = = 0 or = are parabolas turned on their side. = = - v v This curve is not the graph of a function (wh?). 8 Calculus and Structures

10 SECTION. Eample 8: f(,) = = 0 or = Is this the graph of a function? Notice its domain and range on the graph. What are the? = = - Our vocabular of curves now includes: =, =, =, = 3, + =.3 NEW GRAPHS FROM OLD ONES In what follows we will assume that a represents a real number and that a > 0. a)translations i) f( a, ) = 0. This takes the curve and translates it a units to the right. Eample 9: = ( ) = = ( - ) 0 v v Notice that the verte v is translated units to the right and so is the rest of the curve. Calculus and Structures 9

11 LESSON FUNCTIONS Eample 0: = ( ) 3 I I = 3 = ( - ) 3 Notice that the point of inflection I is translated units to the right and so is the rest of the cubic. Eample : ( ) + = 0 c c + = ( ) + = Notice that the center c is translated units to the right and so is the rest of the circle. ii) f(+a,) = 0 This translates a curve a units to the left (remember that a > 0). Use this to sketch the graphs of : =( +), = (+) 3 ; ( + ) + = iii) and iv). f(, a) = 0 and f(, +a)=0. These translate the curve up and down b a units respectivel. Sketch the graphs of : + =, ( + ) =, and + (+) = v) f ( a, b ) combines the above translations so that now a curve can be translated an combination of a and b units to the left, right, up or down. 30 Calculus and Structures

12 SECTION.3 Eample : ( ) + ( + ) = 0 c c (, - ) + = ( ) + ( + ) = Problem 4 : Sketch : a) = (+) ; b) ( + ) = + ; c) ( + ) + ( ) = ; d) + = b. Squeezing or epanding a curve i) f(a,) = 0. This squeezes the curve in the direction of the -ais b a factor of a. Eample 3: = () and = (/) (, ) (, ) (, ) = () = = (/) 3 Notice that the point (,) is squeezed to (/, ), a factor of, in the first curve and epanded to (,) in the second, epanded to a factor of ½, and so is ever other point on these curves. Calculus and Structures 3

13 LESSON FUNCTIONS Eample 4: () + = + = () + = Notice that (,0) is squeezed to (/,0), a factor of, and so are all of the other points. The circle becomes an ellipse. ii) f(, a) = 0. This squeezes the curve in the direction of the -ais b a factor of a. Eample 5: = or = ½ and / = or = (, ) (, ) (, ) / = = = Notice that the point (,) squeezes to (,/) in the first curve and (,) in the second and so do all the other points. Eample 6: sketch the curve: (/) + () = + = 3 (/) + () = c. Reflections i) f(-,) = 0. This reflects a curve as if the -ais was a mirror. 3 Calculus and Structures

14 SECTION.3 Eample 7: = (-) 3 or = - 3 = - 3 = 3 = - 3 = 3 Notice that the cubic reflects across the -ais Eample 8: Consider the curve: = (-) what does = (--) look like? Well = (--) is the same curve as = (+) (wh?) = (-) = (--) = (+) - Notice that the parabola reflects across the -ais But notice that = (-) is the same curve as = so nothing changes. Wh? ii) f(,-) = 0. This reflects the curve as if the -ais is a mirror. Eample 9: Appl this to the curves : = and = 3, i.e., sketch = (or = - ) and = 3 (or = 3 ). Notice that if this rule is applied to + = the circle maintains its position and is unchanged. Wh? iii) f(-,-) = 0. This reflects the curve first in the -ais and then in the -ais. It has the effect of reflecting the curve across the origin in the respect that an point (,) on the original curve maps to the point (-,-) on the transformed curve. Appl this to = 3. This gives = (-) 3 or = 3 so that the curve is unchanged. Calculus and Structures 33

15 LESSON FUNCTIONS Problem 5: For the functions in the graphs below appl the rules of translation, reflections, and squeezing to these transformed functions: a) = f(); b) /3 = f(); c) = f(-); d) = f(-); e) = -f(-) ; f) = f(-) 3 3 f () f () Fig. Fig. 3 3 f () f () Fig. 3 Fig Calculus and Structures

16 SECTION.4.4 SYMMETRIC CURVES If the graph of a function has smmetr, this smmetr can be used to simplif certain math problems as we shall see in Chapters and 3 in the semester. There are two kinds of smmetr that ou should learn about. i. The graph of an function that has the propert, f(-) = f() is smmetric with respect to the - ais, i.e., the -ais acts like a mirror. Such functions are called even. Eample 0: The graphs of f() = or cos are even as shown on the graph below. = -π - π/ π/ π = cos ii. The graph of an function that has the propert, f(-) = -f() is smmetric with respect to the origin of the coordinate sstem, i.e., an value (,) maps to (-,-). Also if ou reflect the curve first in the -ais and then in the -ais its trace is unchanged. Such functions are called odd. Eample : The graphs of f() = 3 and sin are odd as shown on the graph below. = 3 -π - π/ - π/ π = sin Problem 6: Which of the following functions are even, odd, or neither? a) 4 ; b) 3 3 ; c) + 5; d) 3 Calculus and Structures 35

17 LESSON FUNCTIONS.5 COMPLETING THE SQUARE What do the parabolas, = or = look like? We can find out b doing something that mathematicians call completing the square. Here is how it works: a) First consider the case where the coefficient of the term is +, e.g., = b) Group the term and the term together, i.e., ( + ) c) Write these two terms as a square, i.e., + = ( + ). Note that the number in parentheses, +, will alwas be half of the coefficient of the term, ( in this eample) and since the square of the constant term in the parentheses ( in this case) was not there originall ou have to subtract out its square. d) Now add the last number, 4 to get: = = ( + ) +3. e) Rewrite this epression as 3 = ( + ) and draw the graph using the translation rules. What if the coefficient of the term is not +? For eample what does = look like? Follow this procedure: a) Factor out the coefficient of the term, i.e, = -( - + 3) b) Take the term in parentheses and appl the above procedure for the case where the coefficient of is +, i.e, = - ( [ - ] - +3) or = - ( + ) - 4 c) Rewrite the last epression as : + 4 = - ( + ) and use the translation and the squeeze rules of a function to draw the graph. Problem 7: Complete the square for the following parabolas and sketch. a) = ; b) = ; c) 4 + ; d) = COMPOSITE FUNCTIONS Sometimes the output of a function f is placed into the input of another function g as shown in the figure below. Since f() is the output of the f function if is the input, then g(f()) is the output of the g function if f() is the input. The new function g(f()) is called a composite function and sometimes written, g o f. The domain of this function is the domain of the f function and the range is the range of the g function. g o f f f() g g ( f () ) = ( g of ) ( ) 36 Calculus and Structures

18 SECTION.6 Problem 8:. For the following functions f and g where : a) f () =, g() = + ; b) f() = /, g() = 3 + 4; c) f() = 4, g() = find and simplif: f(g()); g(f()); f(g()); g(f()).7 INVERSE FUNCTION Consider the functions: f() = + and g ( ). Check to see that f(g()) = g(f()) =. In this case we sa that g is the inverse function of f and write g f -. It has the propert that, f f f ( ( )) ( f ( )) It is eas to find the formula for an inverse functions. If ou want to find the inverse function of f() = + just set it equal to and solve for, e.g., = + and = ( )/ Then change the back to to get f - () = ( -)/. As we showed in Section., there will be no inverse function f - unless f is one-to-one, in other words for each output from the f function there is onl one input that gives ou that value. Therefore the function must pass the horizontal line test: an horizontal line should cut the graph of t he function no more than once. For eample, f() = + has that propert but f() = does not since f() = f(-) =, e.g., = and = - both give the output value (a horizontal line will cut the graph of = twice). But if we limit the domain of f to onl positive values then f() for 0, does have an inverse. To find the inverse set =, for 0, and solve for : = and = (the positive square root of ) so f - () =. You can check that f f f ( ( )) ( f ( )) Calculus and Structures 37

19 LESSON FUNCTIONS Problem 9: Find the inverse of the following functions: a) f() = 3 + ; b) f() = /; c) f ( ) where..8 ITERATIVE FUNCTIONS Sometimes the output of a function is fed back to itself so that the result is f(f()) as shown in the figure below. This is what is done in a new field of mathematics called chaos theor where a function is fed back to itself over and over again to get f(f(f(f ())))). A whole new theor about science and mathematics has evolved to deal with this possibilit. f o f f f() f f ( f () ) Problem 0: I recentl made a discover about the simple function: f() = -. Take the value =.469 and find f(.469), f(f(.469)), f(f(f(.469))), and f(f(f(f(.469)))). What do ou observe?.9 FUNCTIONS DEFINED BY MORE THAN ONE FORMULA In our calculus and structures course, most of the functions that we will encounter will have to be defined b several formulas because the behave differentl over different intervals. Sometimes these function have values that jump as ou move our finger along the curve. Such curves are said to be discontinuous. 38Calculus and Structures

20 SECTION.9 Eample : An eample of a discontinuous function is a switch that sends a signal s = after time 0 and is off until t = 0 where s = 0 as shown below. Its formula is: s = f(t) = { 0 for t 0 for t > 0 s f ( t ) = f ( t ) = 0 t Note that a circle has been drawn around the endpoint of the portion of the curve t > 0 because the value of the function at t = 0 is 0 not so the point in the circle is not part of the graph Other curves have kinks in them. The simplest eample of such a curve is the absolute value function, f() =. It is defined as follows: So - = while =. = { - if 0 if > 0 f () = - Calculus and Structures 39

21 LESSON FUNCTIONS A function that we will soon encounter in our course is: { 5, 0 6 f() = , , 0 This curve, shown below, is made up of three line segments and it is continuous, the curve has no breaks. Compute: f (), f(6), f() and f(5) Problem : 6 0 Sketch the graph of the discontinuous function: Find: f(3), f(6), f(), f(5) { 5, 0 6 f() = 5, 6-5, 0 40Calculus and Structures

22 Chapter Problems Problems Calculus and Structures 4

23 LESSON FUNCTIONS 4Calculus and Structures

24 Chapter Problems Calculus and Structures 43