UChicago MATH 15200 - FUNCTIONS: A RAPID REVIEW (PART 1)

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Executive review1. What is a function?1.1. Inputs and outputs, or so they say1.2. The real bite: equal inputs give equal outputs1.3. Of circumferences and diameters: an illustrative example2. Some important classes of functions2.1. Constant functions2.2. Polynomial functions2.3. Rational functions3. Computational tools3.1. The domain3.2. The range4. Describing a function4.1. Description by algebraic expression4.2. Description by tabular listing and graphsScaling issues with graphs4.3. The vertical line test, domain and range4.4. Functions defined piecewise4.5. Back to mathematics5. The max and min operatorsFUNCTIONS: A RAPID REVIEW (PART 1)MATH 152, SECTION 55 (VIPUL NAIK)Difficulty level: Easy to moderate. Most of these are ideas you should have encountered either implicitlyor explicitly in the past.Covered in class?: This will roughly correspond to material covered on Monday September 27. Mostof the trickier aspects of this will be covered in class, but many small points will be omitted due to timeconstraints. Hence, it is recommended that you read through these notes either before or after lecture.Corresponding material in the book: Sections 1.5, 1.6. Note that the book covers the same materialwith somewhat different language and different examples of functions, so you should go through it before orwhile doing the homework problems.Corresponding material in homework problems: Homework 1, routine problems 1–6 (all fromsection 1.5), advanced problem 1.Things that students should get immediately: The concepts of function, domain, range, expressionfor function, table of values for a function, graph of a function, the notion of piecewise defined function,polynomials, rational functions, absolute value function, signum function, positive part function.Things that students should get with effort: How to obtain a piecewise definition for a maximumor minimum of two functions, how to determine the domain and range of a function.Executive reviewThis review will probably be reproduced (with minor modifications) in the midterm review sheet. It ismeant as a review summary of these lec ture notes – capturing those aspects of these notes that are importanton a second reading, and ignoring those things that are significant for first time learning but not so importantlater on.For first time reading, skip to the next section.Words ...(1) The domain of a function is the set of possible inputs. The range is the set of possible outputs.When we say f : A → B is a function, we mean that the domain is A, and the range is a subset ofB (possibly equal to B, but also possibly a proper subset).(2) The main fact about functions is that equal inputs give equal outputs. We deal here with functionswhose domain and range are both subsets of the real numbers.(3) We typically define a function using an algebraic expression, e.g. f(x) := 3+sin x. When an algebraicexpression is given without a specified domain, we take the domain to be the largest possible subsetof the real numbers for which the function makes sense.(4) Functions can be defined piecewise, i.e., one definition on one part of the domain, another definitionon another part of the domain. Interesting things happen where the function changes definition.(5) Functions involving absolute values, max of two functions, min of two functions, and other similarconstructions end up having piecewise definitions.Actions (think back to examples where you’ve dealt with these issues)...(1) To find the (maximum possible) domain of a function given using an expression, exclude pointswhere:(a) Any denominator is zero.(b) Any expression under the square root sign is negative.(c) Any expression under the square root sign in the denominator is zero or negative.(2) To find whether a given number a is in the range of a function f, try solving f(x) = a for x in thedomain.1(3) To find the range of a given function f, try solving f (x) = a with a now being an unknown constant.Basically, solve for x in terms of a. The set of a for which there exists one or more value of x solvingthe equation is the range.(4) To write a function defined as H(x) := max{f(x), g(x)} or h(x) := min{f(x), g(x)} using a piecewisedefinition, find the points where f(x) −g(x) is zero, find the points where it is positive, and find thepoints where it is inegative. Accordingly, define h and H on those regions as f or g.(5) To write a function defined as h(x) := |f(x)| piecewise, split into regions based on the sign of f (x).(6) To solve an equation for a function with a piecewise definition, solve for each definition within thepiece (domain) for which that definition is satisfied.1. What is a function?1.1. Inputs and outputs, or so they say. We’re going to begin by talking about functions. You’veprobably already seen functions in some form in calculus and precalculus. You may have seen both thegeneral concept of function and lots of specific examples. In this course, we try to be a lot more preciseabout what a function means. This precision will be very important because functions are used for modelingpurp os es throughout mathematics and mathematically based disciplines.A function is something that “takes in” (or eats or gobbles) an input and “gives out” (or spits) an output.Some people think of a function as a black box or machine into which you feed in input and get output. Forinstance, you put in money into a cola vending machine and get out a cola. In today’s computer age, youmight enter an input value onto a computer screen and get an output. We say that a function maps theinput to the output, so functions are also called mappings or maps. Some people say that a function sendsan input to an output. Functions can also be thought of as rules or assignments.1.2. The real bite: equal inputs give equal outputs. So what’s missing from this description? Well,the most important thing about a function is that when you put in one input, you get one output, and theoutput depends only on the input. In other words, equal inputs should give equal outputs. So it doesn’tdepend on who feeds the input or how the machine is feeling at the time it is fed in. The output dependson the input, and only o n the input. This dependence is what we call a functional dependence.So is Google a function? It takes in your query and outputs a bunch of search results. But in anothersense, it isn’t a function, because Google’s results keep changing with time and other factors. What abouttemperature? Is


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UChicago MATH 15200 - FUNCTIONS: A RAPID REVIEW (PART 1)

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