Math Summar of Important Algebra & Trigonometr Concepts Chapter & Appendi D, Stewart, Calculus Earl Transcendentals Function a rule that assigns to each element in a set D eactl one element, called f ( ), in a set E. Domain the set D is often referred to as the domain of the function. When ou are asked to find the domain of a function, ou will look for -values for which the function is defined. Range the set of all possible values of f ( ) as varies throughout the domain. Functions can be described in four different was.. Verball. Take each real number times itself.. Numericall. 0.5 e π f ( ) 0.5 e 9 π. Visuall 5. Algebraicall f ( ) = In calculus, ou need to be able to work with functions in all four forms. When dealing with an epression, ou need to be able to determine whether or not the epression is a function. You need to reall understand the definition of a function to make this determination. A laperson s definition of a function might be for each -value, there is one and onl one -value. The first eample below is a function and the second one is not. Can ou eplain wh (in both cases)? Eample : F( ).5 0 5 5 Even though we have -values that are the same, each of these -values is associated with two different -values so F( ) is a function. Eample : G( ) 0 5 7 Here we have the same -values being mapped to two different -values so G( ) is not a function. One advantage of the graphical representation of an epression is that it makes it eas to determine whether or not an epression is a function of because ou can use the vertical line test. Created b Dr. Sharon S. Vestal for the SDSU Mathematics & Statistics Department
The Vertical Line Test A curve in the -plane is the graph of as a function of if and onl if no vertical line intersects the curve more than once. Eample : Function Eample : Not a function 5 5 In the graph above, ou can see that no matter where the vertical line is drawn, it will never intersect the curve more than once. In this graph, ou can see that a vertical line will intersect the curve twice. Different kinds of functions There are numerous kinds of functions that ou will encounter in calculus. The ones included in this section are ones that ou should know how to graph and ou should know characteristics of them. An asterisk indicates that the eample is a graph ou should memorize, because the eample is used frequentl in Calculus. Piecewise defined function a function whose definition is different for different portions of its domain. if 0 if < 0 Eample 5: f ( ) = Eample *: f ( ) = = + if > 0 if 0 5 5 Domain: all real numbers or in interval notation, ( ) Range: (,0] (, Range: [ 0, Polnomial function the general form of a polnomial of degree n is n n P( ) = an + an + + a + a0, where an 0. A polnomial of degree 0 is called a constant function and its graph is a horizontal line with equation, = c, where c is the -value where the graph crosses the -ais. Created b Dr. Sharon S. Vestal for the SDSU Mathematics & Statistics Department
A polnomial of degree is called a linear function, with equation given b = f ( ) = m + b, where m 0. A polnomial of degree is called a quadratic function, with equation given b A polnomial of degree is called a cubic function. A polnomial of degree is called a quartic function. Some special polnomials that ou should know are graphed in the following eamples. Eample 7*: f ( ) = Eample 8*: f ( ) = f ( ) = a + b + c, where a 0. 5 5 Range: (, Range: [ 0, Eample 9*: f ( ) = Eample 0*: f ( ) = 5 5 Range: (, Range: [ 0, n / n Root function a root function is a function of the form f ( ) = = =, where n is a positive integer. When finding the domain of the root function, ou should be careful. In Calculus, we will be working with real numbers and since we cannot take an even root of a negative number, the domain for the root function when n is a 0,. positive even integer will be [ ) Created b Dr. Sharon S. Vestal for the SDSU Mathematics & Statistics Department
Eample *: f ( ) = Eample *: f ( ) = 5 5 Domain: [ 0, Range: (, Range: (, P( ) Rational function a rational function is a ratio of two polnomials, f ( ) =. Q( ) You also need to be careful when finding the domain of rational functions. Be sure that ou eclude from the domain -values that would make Q( ) = 0. In addition, ou want to be careful about -values that make both P( ) = 0 and Q( ) = 0 as this results in the indeterminate form, 0. 0 Eample *: f ( ) = (reciprocal function) Eample *: f ( ) = 5 5 0) ( 0, Range: (,0) ( 0, 0) ( 0, Range: ( 0, Algebraic function An algebraic function is a function that is constructed using algebraic operations (such as addition, subtraction, multiplication, division, and taking roots). Polnomials, root functions, and rational functions that have alread been mentioned are eamples of algebraic functions. Some algebraic functions are 5 more complicated, such as f ( ) = + or g( ) =. Trigonometric function Trigonometr is important to the stud of calculus. In addition to knowing the right triangle definitions of the si trigonometric functions, ou should know the unit circle and basic trig identities. You can go to Appendi D in Created b Dr. Sharon S. Vestal for the SDSU Mathematics & Statistics Department
our tetbook to refresh our memor on some trigonometr. In calculus, we alwas use radian measure instead of degree measure when referring to angles. Eample 5*: f ( ) = sin Eample *: f ( ) = csc = csc() π/7 7π/00 9π/0 7π/ 7π/ 7π/ 7π/ 9π/0 7π/00 π/7 9π/00 π/7 7π/00 9π/0 7π/ 7π/ 7π/ 7π/ 9π/0 7π/00 π/7 9π/00 Range: [,] Domain: { kπ, k Z } Range: (, ] [, Eample 7*: f ( ) = cos Eample 8*: f ( ) = sec = sec() π/7 7π/00 9π/0 7π/ 7π/ 7π/ 7π/ 9π/0 7π/00 π/7 9π/00 π/7 7π/00 9π/0 7π/ 7π/ 7π/ 7π/ 9π/0 7π/00 π/7 9π/00 Range: [,] kπ Domain:, k Z,, Range: ( ] [ ) Created b Dr. Sharon S. Vestal for the SDSU Mathematics & Statistics Department 5
Eample 9*: f ( ) = tan Eample 0*: f ( ) = cot 7π/ 7π/ 7π/ 7π/ 9π/0 7π/ 7π/ 7π/ 7π/ 9π/0 kπ Domain:, k Z, Range: ( ) 5 5 Domain: { kπ, k Z } Range: (, Eponential function An eponential function is a function of the form f ( ) = a, where the base a is a positive constant. Some eamples are given below. It is a good idea to review the laws of eponents as these laws are useful when working with eponential functions. Laws of Eponents If a and b are positive numbers and and are an real numbers, then the following are true.. a a = a + a. = a a a. ( ) = a. ( ) ab = a b Eample : f ( ) = = Eample : f ( ) = = = 8 8 7 7 5 5 5 5 Range: ( 0, Range: ( 0, In calculus the most common eponential function that ou will work with is f ( ) = e. This eponential function is used in modeling population growth, radioactive deca, and Newton s Law of Cooling. Created b Dr. Sharon S. Vestal for the SDSU Mathematics & Statistics Department
Eample *: = f ( ) = e Eample *: = f ( ) = e 8 8 7 7 5 5 5 5 Range: ( 0, Range: ( 0, Before we introduce the net tpe of function, let s review the concept of inverse functions. In order for a function to have an inverse that is also a function, the original function must be one-to-one. One-to-one A function f is called a one-to-one function if it never takes on the same value twice (i.e. for each output value (-value) there is onl one input value (-value). One eas wa to check to see if a function is one-to-one is to use the horizontal line test. Horizontal Line Test A function is one-to-one if and onl if no horizontal line intersects its graph more than once. Eample 5: f ( ) = Not one-to-one so its inverse is not a function Eample : f ( ) = One-to-one so its inverse is a function. 5 5 Created b Dr. Sharon S. Vestal for the SDSU Mathematics & Statistics Department 7
The graph of the inverse mapping of f ( ) = is shown here. Clearl, it is not a function because it doesn t pass the vertical line test. The graph of the inverse of f ( ) = is shown here. Notice that it will pass the vertical line test so it is a function. 5 5 Notice that the function and the inverse function in Eample are familiar. The function f ( ) = has an inverse of f ( ) =. The notation f denotes the inverse of the function f. It does not mean the reciprocal of f. One observation that ou might make from the graphs in Eample is that it looks like one graph can be obtained b the other graph b flipping the graph across the line =. This is illustrated in the pictures below. Visual representation of f = ( ). f ( ) = and its inverse Visual representation of f ( ) = e and its inverse f ( ) = ln. 5 5 When ou compose a function and its inverse, ou alwas get for all in the domain of the inside function. In other f f ( ) = f f ( ) = for all in the domain of f. words, ( ) for all in the domain of f. Likewise, ( ) For some functions, it is not possible to find an inverse using algebraic means, even though one eists. Therefore, we give the inverse function a name and define it in terms of the original function. For eample, f ( ) = log a is the inverse of the function f ( ) = a. We sa that log a = when a =. Logarithmic functions A logarithmic function is a function of the form f ( ) = log a, where the base a is a positive constant. Logarithmic functions are inverses of eponential functions. There are two tpes of logarithmic functions that are dealt with frequentl. The common logarithm is defined as f ( ) = log. On our calculator, the LOG ke refers to the common logarithm. The natural logarithm is used 0 Created b Dr. Sharon S. Vestal for the SDSU Mathematics & Statistics Department 8
more in calculus (and higher level mathematics). It is defined as f ( ) = loge = ln. On our calculator, the LN ke refers to the natural logarithm. The graph of f ( ) = ln. is shown below and it should not be surprising since it is the inverse function of = e. Eample 7*: f ( ) = ln 5 We know that Eample and Eample 7 are inverses of each other. Some observations to notice are that the domain for Eample is the range for Eample 7; and the range for Eample is the domain for Eample 7. This is not a coincidence it follows from the fact that the are inverses of each other. Domain: ( 0, Range: (, Since we are talking about logarithmic functions, it might be a good time to review some properties of logarithms. Properties of Logarithms Given a,, and positive numbers, then the following are true. log = log + log. ( ) a a a. loga = loga loga r log = r log, where r is an real number.. a ( ). ( ) 5. ln e =, R ln e =, > 0 a Properties and 5 follow directl from the fact that f ( ) = e and f ( ) = ln are inverses of each other. Transcendental Functions Transcendental functions are functions that are not algebraic. Some eamples of transcendental functions include the trigonometric functions, eponential functions, logarithmic functions, and inverse trig functions (which we will eplore in detail later). Transformations of Functions Now that we have encountered various tpes of functions, let s discuss how we can transform functions. Understanding these transformations and knowing the graphs that ou have seen throughout this document, ou should be able to graph lots of functions ver quickl. Vertical and Horizontal Shifts. Suppose c > 0. To obtain the graph of = f ( ) + c, shift the graph of = f ( ) a distance of c units upward. = f ( ) c, shift the graph of = f ( ) a distance of c units downward. = f ( c), shift the graph of = f ( ) a distance of c units to the right. = f ( + c), shift the graph of = f ( ) a distance of c units to the left. Created b Dr. Sharon S. Vestal for the SDSU Mathematics & Statistics Department 9
Eample 8: = + Eample 9: = 5 5 Eample 0: = Eample : = + 5 7 5 Reflections. To obtain the graph of = f ( ), reflect the graph of = f ( ) about the -ais. = f ( ), reflect the graph of = f ( ) about the -ais. Eample : = Eample : = 5 5 Vertical and Horizontal Stretching. Suppose c >. To obtain the graph of = cf ( ), stretch the graph of = f ( ) verticall b a factor of c. = f ( ), compress the graph of = f ( ) verticall b a factor of c. c = f ( c), compress the graph of = f ( ) horizontall b a factor of c. Created b Dr. Sharon S. Vestal for the SDSU Mathematics & Statistics Department 0
= f, c Eample : = 7 stretch the graph of = f ( ) horizontall b a factor of c. Eample 5: = 7 5 5 5 5 Eample : = sin( ) Eample 7: = sin 5 5 5 5 Combinations of Functions We have alread briefl discussed combining functions, using the composition of functions. We will discuss composition again as well as other operations that can be used to combine functions. Two functions f and g can be combined to form new functions in a manner similar to the wa we add, subtract, multipl, and divide real numbers.. ( f + g )( ) = f ( ) + g( ). ( f g )( ) = f ( ) g( ) f f ( ). ( f g )( ) = f ( ) g( ). ( ) =, g( ) 0 g g( ) One important use of combining function in calculus is simplifing the difference quotient. f ( + h) f ( ) The difference quotient is defined as. h This represents the slope of the secant line of a curve, as seen in the picture at right. Recall, that the slope of the line is represented b the change in divided b the change in. In the picture, we find the slope of the secant line f ( + h) f ( ) f ( + h) f ( ) b evaluating m = = =. ( + h) h (, f()) (+h, f(+h)) =f() secant line Created b Dr. Sharon S. Vestal for the SDSU Mathematics & Statistics Department
Understanding the difference quotient is vital in the stud of calculus. Eample 8: Simplif the difference quotient for f ( ) =. f ( + h) f ( ) = + h h h ( + h) ( + h) = h h ( + h) = h h ( + h) = h h = ( + h) h =. ( + h) Let s discuss the composition of functions again. We discussed this concept earlier when talking about a function and its inverse function. Now we will talk about composition of functions more in depth. Composition of f and g Given two functions f and g, the composite function f g is defined b f g = f g( ). The domain of f g is the set of all in the domain of g such that g( ) is in the domain of f. ( )( ) ( ) In other words, ( f g )( ) is defined whenever both g( ) and f ( g( )) are defined. Composition is important in calculus because often we have to decompose a function into its parts. For eample, given F( ) = cos, find f ( ) and ( ) F ( ) = f g ( ). In this case, we know that g has to be the inside g so that ( ) function so g( ) = cos, and f has to be the outside function so f ( ) =. To check our answer, we look at This confirms that our answer is correct. Focus on Trigonometr ( f g) ( ) = f ( g( )) = f (cos ) = cos = F( ). Trigonometr is an essential part of the stud of calculus. A good trigonometr background is essential for ou as a student of calculus. This section will focus on some concepts from trigonometr that ou need to know in order for ou to be successful in the calculus sequence. Do not think of this section as a series of facts to memorize but rather information that ou need to learn and understand. Can ou answer the following questions without having to look at a tetbook? If not, then ou need to spend a significant amount of time on this section and with Appendi D in our tetbook. Created b Dr. Sharon S. Vestal for the SDSU Mathematics & Statistics Department
π. sin =. cos () =. The most famous of the Pthagorean Identities is. Answers:. ½. 0. The Unit Circle sin + cos = You need to know the values of all si trigonometric functions at an of the special angles on the unit circle. If ou can learn these values for the angles in quadrant I, then ou should be able to translate this knowledge to an of the important angles in the other quadrants. Essentiall, ou need to know the Pthagorean Theorem and some geometr and ou must learn the unit circle. If ou feel that ou alread have a good grasp of the unit circle, but just need a quick refresher, skip to page 5. Thorough Eplanation for Cosine and Sine of π Looking at the picture to the right, ou see a right isosceles triangle. In fact, we have a 5-5-90 triangle. Using the Pthagorean Theorem, we get the equation + =. Solving this equation for ields = ± and since the point is in the quadrant I, we know that must be positive. So, we have that the length of each leg of this right triangle is or. The coordinates of the point on the circle are,. When using the unit circle, the cosine of the angle between the hpotenuse and the positive -ais is given b the -coordinate and the sine of the angle is given b the -coordinate. In calculus, we alwas use radians instead of degrees so we have that π π cos = and sin =. 0.5 - -0.5 - Understanding this information and knowing the sign of and in all of the quadrants allows us to know the values of the trigonometric functions for the following angles on the unit circle: π, 5π, and 7π. Created b Dr. Sharon S. Vestal for the SDSU Mathematics & Statistics Department
π π Thorough Eplanation for Cosine and Sine of and The triangle ACB was constructed as an equilateral triangle. Since BA and BC are radii of the unit circle, the have length one unit. Thus, the length of AC will also be one unit. From geometr, we know that an equilateral triangle is also equiangular so the measures of the angles at A, C, and B are all 0 degrees or π radians. BD was constructed as the perpendicular bisector of AC, and we know from geometr that it will also be the angle bisector for angle B. Thus, we have that triangle BAD is a 0-0- 90 triangle with BA =, and AD = so b the Pthagorean Theorem we can solve for BD..5 0.5 A D - B -0.5 C - BD + = will give us that BD = ± a length on the positive -ais, we have that, we have that remember that BD =, but since BD is. Now, we have the coordinates of the point A and the are. Since the -coordinate represents the cosine of ABD and the -coordinate represents the sine of ABD, π π cos = and sin =. π ABC was and BD is the angle bisector so ABD π π Now ou might be wondering wh it is instead of, but π will be half of the angle,. In triangle ABD, the angle at A is 0 degrees or π radians and if we recall that the cosine of an angle in a right triangle is defined as adjacent hpotenuse and that the sine of the angle in a right triangle is defined as opposite, hpotenuse cosine and sine of π. Using triangle ABD with angle A equal to π radians, we have that -.5 we can use these facts to find the π adjacent / cos = = = hpotenuse and π opposite / sin = = =. hpotenuse Understanding this information and knowing the sign of and in all of the quadrants allows us to know the values of the 5 7 5 trigonometric functions for the following angles on the unit circle: π, π, π, π, π, and π. Now we have the information needed to find the values of all si trigonometric functions at all the special angles given on the unit circle on the net page. Created b Dr. Sharon S. Vestal for the SDSU Mathematics & Statistics Department
o 0 ; o π 5 ; o 5π 50 ; π 90 o ; π ; ( 0,) 0 o ; π ;, π o 5 ; ; 0 o ; π ;,, o 80 ; π ; (,0 ) 0 o or 0 0 or π ; o (,0 ) o 7π o π 0 ; 0 ; o 5π o 7π 5 ; 5 ; o π 0 ; o 5π 00 ; o π 70 ; ; ( 0, ) Recall that the unit circle is a circle centered at the origin of radius so its equation will be given b + =. Using the equation of the unit circle and the fact that the cosine of an angle is given b the -value at the point on the unit circle and the sine of the angle is given b the -value of the point on the unit circle, we can derive the most important of the three Pthagorean Identities + = ( ) ( ) cosα + sinα = or cos α + sin α =. Now to derive the other two Pthagorean Identities, we divide the above equation b This gives us the following cos α sin α + = + tan α = sec α cos α cos α cos α and α α + = α + = sin α sin α sin α cos sin cot csc. cos α and α sin α, respectivel. Some other trigonometric identities that ou should know to help ou in our stud of calculus include the following. Addition and Subtraction Formulas: sin α ± β = sinα cos β ± cosα sin β ( ) ( ) cos α ± β = cosα cos β sinα sin β From these two formulas, we can derive the Double-Angle Formulas: sin α = sinα cos α and cos α = cos α sin α = α cos α sin. = Created b Dr. Sharon S. Vestal for the SDSU Mathematics & Statistics Department 5
From the second double-angle formula, ou can derive the Half-Angle Formulas: cos α + cos α sin α = and cos α =. Inverse Trigonometric Functions Hopefull b now ou are becoming more comfortable with trigonometr. Recall the graphs of the si trigonometric functions on pages and 7. We know that these functions are periodic (i.e. the graph repeats itself after a certain period of time). So, how can these trigonometric functions have inverse functions? We have a problem because we know that for a function to have an inverse function, the original function must be one-to-one and none of these graphs will pass the horizontal line test. So, in order to discuss inverse trigonometric functions, we must take the original si trigonometric functions and restrict their domains so that the are one-to-one and we can talk about their inverse. Let's look at the sine curve first and how we should restrict the domain. As seen in the graph on the right, it seems that if we restrict π π the domain of the sine function to the interval,, then,. we can still get the entire range of the function, [ ] Thus, we have that the sine function has an inverse function with this restricted domain. Also, we know that the domain and range of the inverse function correspond to the range and domain of the original function. The notation that we will use for the inverse function of f ( ) = sin is f ( ) = sin ( ). The graph is shown below in the darker color with the graph of = sin with restricted domain shown in the lighter color. Looking at these graphs, ou can see that the are inverses of each other because the line = acts as a mirror for the two graphs. Eample 9*: = sin ( ) 7π/00 7π/ 7π/ 7π/00 9π/00 7π / 7π/ 7π/ 7π/ 7π/ Domain: [,] 7π/ π π ; Range:, This same process can be done for all si trigonometric functions so that the will be one-to-one and have inverse functions. The table on the following page summarizes the si trigonometric functions with restricted domains, and their inverse functions with their domain and range. The domain and range for sec are not standard in all tetbooks so ou ma see this different somewhere else. Created b Dr. Sharon S. Vestal for the SDSU Mathematics & Statistics Department
Function Restricted Domain π π sin, cos [,π ] tan π π, csc π,0 0, sec π π 0,, π cot (,π ) Range Inverse Function Domain Range [, ] 0 [, ] (, ) (, ] [, ) (, ] [, ) 0 (, ) sin ( ) or arcsin( ) cos ( ) or arccos( ) tan ( ) or arctan( ) csc ( ) or arccsc( ) sec ( ) or arcsec( ) cot ( ) or arccot( ) Let's look at two other important inverse trigonometric functions and their graphs. π π, [, ] [, ] [ 0,π ] (, ) π π,,, π,0 0,,, π π 0,, π ( ] [ ) ( ] [ ) (, ) ( 0,π ) Eample 0*: = cos ( ) Eample *: = tan ( ) 9π/0 7π/00 =arccos() 7π/ 9π/0 7π/ 7π/ = arctan() 7π/ 9π/0 7π/ 7π/ 7π/ 7π/ 9π/0 7π/00 9π/0 7π/ 7π/ 7π/ 7π/ 9π/0 7π/00 π 7π/ 7π/ 7π/ 7π/ 9π/0 Domain: [, ] Range: [ 0,π ] 9π/0 Domain: (, ) Range: π π, 7π/00 π/7 Now let's work through a couple of problems involving inverse trigonometric functions so ou feel more confident in our knowledge. π Eample : Find the eact value of the epression, sin. This problem involves knowing our unit circle. Essentiall we need to find the -coordinate of the point corresponding to the angle,. π So, our answer needs to be a number between and because that is the range of the sine function. Looking back at the unit circle on page 5, we see that the -coordinate is. Now this is something that ou should Created b Dr. Sharon S. Vestal for the SDSU Mathematics & Statistics Department 7
not have to look up ou should know this answer without having to look at the unit circle. Thus, we have that π sin =. In the net eample, ou are given information about the point on the unit circle and ou need to find the angle that corresponds to this information. It is ver important to pa attention to the range of the inverse trigonometric functions sin = means that sin( ) =. because it tells ou which angles are allowed for our answers. The statement that ( ) For a trigonometric function, the input value is an angle and the output is a number, while for an inverse trigonometric function, the input is a number and the output is an angle. Eample : Find the eact value of each epression. a) sin Let's reword this problem so we can understand what is being asked. It is asking ou to find the angle on the unit circle that has a sine (-value) equal to -/. If ou go back and look at our unit circle, ou realize that there is more than one answer. This is where the range of the inverse sine function becomes ver important. The angle that we give as an π answer must lie in this range. Now, ou can see that there is onl one answer and it is. Thus we have that sin π =. tan b) ( ) This question is asking us to find the angle on the unit circle that has a tangent equal to. It might be easier for us to sinα think about this in terms of sine and cosine. Recall that tanα =, so the angle that we are looking for must be the cosα angle α where sin α =. This means that sin cos, cosα α = α and since we know from our unit circle that the - and - coordinates of the point must be equal. You will notice that this occurs more than once, but remember that our answer π has to lie in the range of the inverse tangent function. So, we can see that this happens when α =. Thus we have that π tan ( ) =. Essentiall, the material covered in this packet is essential algebra and trigonometr skills that ou need to be successful in calculus. If ou are finding that our skills are etremel rust in these areas, ou should talk with our Math instructor for suggestions on how to be a successful student of calculus. More review material can be found on the Gatewa Eam website, http://teach.sdstate.edu/users/vestals/gatewa.htm ; and for a fee of $0, ou can go to www.mmathtest.com and take a diagnostic test, which will identif what skills ou need to practice and allow ou to do practice problems. Created b Dr. Sharon S. Vestal for the SDSU Mathematics & Statistics Department 8