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Notes on Part 1, Functions Elementary Functions Chapter 1, Functions c Ken W. Smith, 2013 Version 1.3, January 8, 2014 These notes were developed by professor Ken W. Smith for MATH 1410 sections at Sam Houston State University, Huntsville, TX. This material was covered in six 80-minute class lectures at Sam Houston in Summer 2013. (Sections 1.0 and 1.1 were combined in one lecture since 1.0 is a brief review.) In addition to these class notes, there is a slide presentation version of these notes and a set of Worksheets. All of these are available on Blackboard. Contents 1 Functions and Polynomials 3 1.0 Algebra excellence . .3 1.0.1 Exponential notation { merely an abbreviation! . .3 1.0.2 Kilobytes and powers of ten (an application to computer science) . .4 1.0.3 Polynomial arithmetic: a review of A2 − B2 and other basic factoring . .5 1.0.4 Simplifying complicated fractions . .6 1.0.5 Other resources for algebra review...........................7 1.1 An introduction to functions . .8 1.1.1 The function machine . .8 1.1.2 Functions as ordered pairs . 10 1.1.3 Functions defined by equations . 12 1.1.4 The domain of a function . 14 1.1.5 Other resources for functions and function notation................. 15 1.2 Graphs of functions . 17 1.2.1 Using ordered pairs to draw functions . 17 1.2.2 Intercepts of the graph of a function . 19 1.2.3 Intervals in which the function rises or falls . 21 1.2.4 The vertical line test definition of a function . 22 1.2.5 Piecewise functions . 22 1.2.6 Other resources for graphing functions......................... 26 1.3 Transformations of functions . 27 1.3.1 Vertical shifts . 27 1.3.2 Horizontal shifts . 28 1.3.3 Vertical expansions . 29 1.3.4 Horizontal expansions . 29 1.3.5 Combining these expansions . 30 1.3.6 Resources for function transformations......................... 33 1.4 Symmetries of functions . 34 1.4.1 Even and odd functions . 34 1.4.2 Periodic functions . 37 1.4.3 The greatest integer function and other interesting examples . 38 1.4.4 Resources for function symmetries........................... 39 1.5 Function composition . 41 1.5.1 The algebra of functions . 41 1.5.2 Composing two functions . 41 1.5.3 Order is important! (f ◦ g 6= g ◦ f!)........................... 43 1.5.4 Chaining functions together . 44 1.5.5 Resources for the composition of functions....................... 45 1 1.6 Inverse functions . 46 1.6.1 The concept of inverse function . 46 1.6.2 Changing the domain in order to create an inverse function. 47 1.6.3 Finding the inverse of a function . 48 1.6.4 The geometric meaning of inverse (and the horizontal line test) . 50 1.6.5 The mathematical meaning of \inverse" . 50 1.6.6 Requirements for the existence of an inverse function* . 51 1.6.7 Resources for inverse functions............................. 52 2 1 Functions and Polynomials 1.0 Algebra excellence Before we study the elementary functions of science and calculus, we need some comfort with algebra. In this brief lecture we review two important ideas: (1) computations using exponential notation and (2) operations of polynomial arithmetic. These are two major algebra computations we will do throughout this class (and scientists will use throughout their careers!) This review is brief and is not intended to be comprehensive. Our precalculus class will assume that students are comfortable with most of the major concepts of elementary and intermediate algebra and we will not, in general, review those concepts in this class. 1.0.1 Exponential notation { merely an abbreviation! We abbreviate x · x · x by x3. This notation (merely an abbreviation!) quickly leads to some rules on how one should treat exponents. For example, since x3 · x2 = (x · x · x) · (x · x) = x5 then when we multiply objects with the same base (x) we should add the exponents: xmxn = xm+n. (1) Similarly, x3 x · x · x x x x = = = x; x2 x · x 1 x x so when we divide objects with the same base (x) we should subtract the exponents: xm = xm−n. (2) xn What if we use exponents in sequence, that is, we raise x to a power and then raise that result to a second power? For example, (x3)2 = (x · x · x)2 = (x · x · x)(x · x · x) = x · x · x · x · x · x = x6: Here our \abbreviation" leads us to multiplying exponents. We may generalize from this that (xm)n = xmn. (3) Repeated exponentiation leads to multiplying exponents. Our understanding of the exponent \abbreviation" has quickly led us to three natural rules about manipulating exponents. These algebra \rules" are merely the effects of the algebraic symbolism. There are other effects of our algebraic symbolism. Once we get used to the impact of this notation, we see that since multiplying by 1 leaves a number unchanged and since multiplication by x0 also leaves a number unchanged (xnx0 = xn+0 = xn) then 1 and x0 must be the same: x0 = 1. (4) 3 We can extend our exponent notation to rational exponents. Since 1 2 1 ·2 1 (x 2 ) = x 2 = x = x and since p ( x)2 = x 1 p then x 2 must represent x: More generally, denominators in exponents represent roots: 1 p x q = q x. (5) Some examples. First we practice our understanding of exponentiation: 2 1. Simplify 8 3 : 2 1 We solve this by recognizing that the fractional exponent 3 represents ( 3 )(2), so we will take a 1 cube root (that is the meaning of the exponent 3 ) and then we will square the result. Solution. p 2 1=3 2 3 2 2 8 3 = (8 ) = ( 8) = 2 = 4: Here is another, similar example. 3 2. Simplify 4 2 : Solution. p 3 1=2 3 3 3 4 2 = (4 ) = ( 4) = 2 = 8 1.0.2 Kilobytes and powers of ten (an application to computer science) Here is an application appearing in a number of computer science computations. We note that 210 = 1024 while 1000 = 103: The computer scientist works with computer registers which use bits (zeroes and ones) and so storage and memory are measured in powers of two. (We say that computer science computations are done in base two.) Yet the language of computer science is often based on our traditional powers of ten, where Greek prefixes such as kilo- represent a thousand, mega- represents a million and giga- a billion. However, to a computer scientist, the prefix kilo- really represents 210, not 103: A kilobyte is 210 = 1024 bytes; a gigabyte is 230 bytes. Let us approximate 230 as a power of ten: Since 230 = (210)3 and since we approximate 210 or 103 then 230 = (210)3 ≈ (103)3 = 109: Exercise. How many digits are there in 2300? Solution. Write 2300 = (210)30 ≈ (103)30 = 1090: Now 1090 = 1 × 1090 is 1 followed by 90 zeroes so 1090 has 91 digits. Therefore 2300 should have 91 digits. (A detour to WolframAlpha and a quick computation indeed gives 2300 = 2037035976334486086268445688409378161051468393665936250636140449354381299763336706183397376 You can check that this has 91 digits! But since 210 > 103, it turns out that 2300 is more closely approximated by 2 × 1090 than 1 × 1090 .) 4 Practice. Here are some sample problems from an old precalculus quiz. Simplify the following expressions: p 3 x6 1. p x2 1 (x6) 3 x−2 2. x4 Solutions. p 1. We follow the meaning of the exponent, rewriting expressions such as 3 x6 as x2 since (x2)3 = x6: So p 3 x6 x2 p = = x: x2 x 2 6 1 2 2 −2 x 2. We first simplify the numerator, noting that (x ) 3 = x and x x = = 1: So x2 1 (x6) 3 x−2 x2x−2 1 = = or x−4 x4 x4 x4 1.0.3 Polynomial arithmetic: a review of A2 − B2 and other basic factoring There are some basic polynomial expansion concepts that will appear throughout precalculus, calculus, and computations in the sciences. For example, at one point we learned to use the distributive law to expand (\FOIL") expressions like: (x + 5)(x − 3) = x2 − 3x + 5x − 15 = x2 + 2x − 15 and then to factor expressions like x2 + 2x − 15 by reversing this process. If we expand the expression (A + B)2 = (A + B)(A + B) we discover, in addition to the obvious squares A2 and B2 the \cross term" 2AB: However, if instead we expand (A + B)(A − B) we obtain A2 − B2; the cross term involved both AB and −AB and these cancelled out. We will use these basic patterns repeatedly in this course: (A + B)2 = A2 + 2AB + B2 (6) and (A + B)(A − B) = A2 − B2 (7) In the first case (equation 6), notice the existence of a middle term caused by our polynomial expansion. Don't make the \freshman" mistake of thinking that (A + B)2 is just the sum of the squares of A and B! In the second case (equation 7), we see that the difference of two squares nicely factors into the product of the sum and difference of the elements. Examples. Here are some other simplification problems. Each involves factoring of some type. Notice that the expression we are factoring in problems 2 and 3 have the same pattern as problem 1; recognizing that pattern leads us to our solution. Simplify: 5 x2 − 9 1. x + 3 x4 − 9 2. x2 + 3 x − 9 3. p x + 3 Solutions. x2 − 9 (x − 3)(x + 3) 1. = = x − 3: x + 3 x + 3 x4 − 9 (x2 − 3)(x2 + 3) 2.
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