Section 1.: A Catalog of Functions As we discussed in the last section, in the sciences, we often tr to find an equation which models some given phenomenon in the real world - for eample, temperature as a time of da or velocit in terms of time. Such a model is called a mathematical model. In this section we shall consider some of the more common models used which ou should have seen in previous classes. 1. Linear Functions A linear function is a function whose graph is a line (non-vertical). Equivalentl, a linear function is a function with equation f() = m + b where m is called the slope of the function. Such functions are eas to classif - once we know the slope and a single point on the line, or two points on the line, then we can write down an equation for the function. This means that linear models in the real world are arguabl the easiest ones to work with. We shall summarize some of the facts about linear functions. The following gives us different was to write down the equation for a linear function. Result 1.1. (i) (Slope-Intercept form) If a linear function has slope m and -intercept (0, b), then it has equation f() = m + b. (ii) (Point-Slope form) If a linear function has slope m and passed through the point (a, b), then it has equation f() = m( a) + b. Of these two formulas, point-slope form is probabl the most useful since it does not require knowing the intercept (which sometimes we will not be given). If we are given two points ( 1, 1 ) and (, ) instead of a point and a slope, we simpl calculate the slope b calculating rise over run: = 1. 1 We finish b looking at a couple of eamples. Eample 1.. Find a equations for the famil of linear functions whose slope is. Since the onl condition is that the slope is, an such function will be of the form f() = + b where b varies over all real numbers. Eample 1.3. Does ever line define a linear function? No - vertical lines do not because the do not pass the vertical line test. 1
Eample 1.. Find the equation for the linear function which passes through the points (1, ) and (e, π). First, the slope is = π e 1. Using point-slope form with the point (1, ), we get ( ) π = ( 1) +. e 1 Alternativel, we could have had ( ) π = ( e) + π. e 1. Power Functions and Polnomials The net simplest tpe of equation is the polnomial. In order to full understand polnomials however, we shall first consider power functions..1. Power Functions. We start with the definition of a power function. Definition.1. A function of the form f() = a n for n an integer with n 1 and a an real number is called a power function. Clearl the domain of all power functions is all real numbers. To determine the ranges, we can look at the different graphs. There are 6 basic shapes for a power function depending upon whether a is positive or negative and whether n is odd, even or 1. Since we have alread considered n = 1 (linear functions), we instead consider n. n odd even 0 0 3 0 1 a positive 0 K K1 0 1 Range: (, ) Range: [0, ) 0 0 K1 0 negative K0 0 K Range: (, ) Range: (, 0]
We can also consider other tpes of power function where n is a fraction or n is a negative number. These graphs and functions however are more complicated, so we just consider two more eplicit eamples. Eample.. If we consider negative integers, so power functions of the form f() = a n where a is an nonzero real number and n is a negative integer, we also get four graphs. In all cases, the domain is all reals ecept = 0, so the set (, 0) (0, ). The ranges are given in the table below. n odd even 3 K1.0 K0. 0 0. 1.0 K1.0 K0. 0 0. 1.0 a positive Range: (, 0) (0, ) Range: (0, ) K1.0 K0. 0 0. 1.0 K1.0 K0. 0 0. 1.0 negative Range: (, 0) (0, ) Range: (, 0) Eample.3. Fractional powers are much more difficult, so instead we just consider power functions of the form f() = a 1 n where n is a positive integer (we sometimes call these root functions). Again, we get four different graphs. The main difference in this case however is that the domains are different. n odd even a positive K K Domain: (, ) Domain: (0, ) Range: (, ) Range: (0, ) K K negative K K Domain: (, ) Domain: (0, ) Range: (, ) Range: (, 0) K K
.. Polnomial Functions. More general than power functions are polnomials defined as follows: Definition.. A polnomial is a function with equation p() = a 0 + a 1 + a +... a n n where the a i are integers and a n 0. For a polnomial as given in the definition, we use the following terminolog and have the following facts: (i) n is the degree of the polnomial (the largest power with nonzero coefficient). (ii) a n n is called the leading term and a n is called the leading coefficient. (iii) The domain is all real numbers. (iv) If n is odd, the range is all real numbers. If n is even, and a n is negative, there is a number N such that the range is (, N] and if n is even, and a n is positive, there is a number N such that the range is [N, ) One ver useful fact sas that though we ma not be able to completel determine what a polnomial looks like everwhere, we can alwas work out its end behavior: Result.. (The Leading Term Propert) Suppose p() is a polnomial and a n n is its leading term. Then the end behaviour of p() and a n n are the same. Unlike power functions, we cannot determine all the information about polnomials in general without being given an eplicit polnomial. There are some interesting facts which can be derived from just looking at its degree however. We finish with some eamples. Eample.6. Eplain the following facts: (i) If the degree of p() is n, there are at most n zeros (ii) If p() has even degree, it can have anwhere between 0 and n zeros (iii) If p() has odd degree, it can have anwhere between 1 and n zeros (iv) If p() has even degree, it can have anwhere between 1 and n 1 turning points (v) If p() has odd degree, it can have anwhere between 0 and n 1 turning points Eample.7. How do ou know the following is not the complete graph of a degree 9 polnomial? Could it be the complete graph of a degree 6 polnomial?
30 0 K0 K30 The end behavior does not match a degree 9 power function. However, it does match the end behavior of a degree 6 power function, so it could conceivabl be a degree 6 polnomial. Eample.8. Find a possible formula for the polnomial graphed below. 0 30 0 K0 K30 K0 This polnomial has a zero of odd degree at = 3 and = and a zero of even degree at = 1. Therefore a possible formula would be p() = a( + 3)( 1) ( ) where a is some positive real number. Net note that the graph passes through (0, 6), so plugging in those coordinates, we get a = 1, so p() = ( + 3)( 1) ( ). 3. Rational Functions Though the are built up from polnomials, rational functions are much more complicated. The formal definition of a rational function is as follows: Definition 3.1. A rational function is a quotient of polnomials, R() = p() q() where p() and q() are polnomials. The domain of a rational function are all points where q() 0.
6 If a point a is not in the domain of R(), then q(a) = 0 which means that ( a) divides q(). If ( a) n is the largest power of a dividing q(), we sa that a is a zero of q() of order n. If a is a zero of q() of order n, there are two possible things which can happen at = a: (i) If p() has a smaller degree zero at = a, then there is a vertical asmptote at = a (ii) If p() has an equal or larger degree zero at = a, then there is a hole in the graph at = a. Similar to polnomials, the end behaviour of a rational function can also be determined b leading coefficients. Specificall, we have the following: Result 3.. (The Leading Term Propert for Rational Functions) If R() = p()/q() and a n n and b m m are the leading terms of p and q respectivel, then the end behavior of R() is the same as that of a n n /b m m. Besides these observations, there is little else which can be said about rational functions without looking at eplicit eamples - not even the range. We shall eplore rational functions again later, but we consider an eample before we do. Eample 3.3. Find the domain of and sketch its graph. R() = 3 1 1 First, the domain will be all real numbers ecept the zeros of 1, so ±1. In interval notation, (, 1) ( 1, 1) (1, ). Now notice that 3 1 = ( 1)( + + 1) and 1 = ( 1)( + 1, so the graph of R() will be the same as that of + + 1 + 1 with a hole at = 1. Finall note that there will be an asmptote at = 1, there are no zeros since the numerator has no zeros, and the end behavior is similar to / =. Thus the graph must look something like the following:
7. Algebraic Functions Definition.1. An algebraic function is a function which can be constructed from polnomials using the basic algebraic operations (addition, subtraction, multiplication, taking powers and roots). Algebraic functions are much too numerous to attempt to classif, so instead we give a couple of eamples. Eample.. The following are all algebraic functions: (i) p() = ( + 1) (ii) q() = + 1 1 The graphs of such functions can be almost anthing, as can domains/ranges etc. Eample.3. Construct an algebraic function with domain (, ). Consider the function R() = 1 1 +. + Clearl we cannot have = ±. Also, since square roots cannot be negative, we must have 0 or and + 0 or, so putting these two together, we have < <.. Trigonometric Functions Before we even consider trigonometric functions, we make the ver important observation that degrees do not eist - in calculus, we alwas use radians. All trig functions can be built using basic algebraic operations from the two trig functions sin () and cos (). Therefore, we shall first recall some basic facts about these two trig functions. (i) The domain of sin () and cos () is all real numbers. (ii) The domains of sin () and cos () are all numbers between 1 and 1 inclusive, so [ 1, 1]. (iii) sin () and cos () are both periodic with period π. This means sin ( + nπ = sin ()) and cos ( + nπ) = cos () for an integer n. (iv) sin () and cos () satisf the ver important identit sin () + cos () = 1. (v) The zeros of sin () are at the integer multiples of π. (vi) The zeros of cos () are at the non integer multiples of π/.
8 (vii) The graph of sin () is 1.0 0. K0. (viii) The graph of cos () is K1.0 1.0 0. K0. K1.0 The other important trigonometric functions are tan() = sin ()/ cos (), cot() = 1/ tan(), sec () = 1/ cos() and csc () = 1/ sin(). 6. Eponential Functions Definition 6.1. An eponential function is a function of the form f() = a where a > 0 is some real number. We postpone a treatment of eponential functions until Section 1.. 7. Logarithmic Functioins Definition 7.1. If f() = a is an eponential function, then the logarithm base a, denoted log a () is defined to be the inverse function of f(). We postpone a treatment of logarithmic functions until Section 1.6. 8. Transcedental Functions Definition 8.1. We define a transcendental function to ba an function which is not algebraic. We finish with an eample. Eample 8.. Classif the following functions: (i) f() = 1. (ii) f() = 9 +. (iii) f() = 1. (iv) f() = sin (). (v) f() =.
1 3 are all algebraic and are transcendental. More specificall, 1 is a power (or root) function and is a polnomial. 9