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Math 110 Chapter 2, Section 2 Page 1 / 3 Review: Functions (§2.1) The goal here is to review what they should already know about functions. I think that the idea of a function's "domain" won't really have been stressed much prior to this course, so put some emphasis on that. Pictures: Determining if a mapping represents a function <Show the pictures below, and go over the following definitions> Def: Relation: Correspondence between 2 sets (XY) Def: Function: A relation that associates each element in the domain with EXACTLY ONE element of the range. Def: Domain: A non-empty set which is mapped to another set by a function Def: Range: A non-empty set which is mapped to another set by a function Students (the domain) Bob Jo Mary Joe Sarah Rob Jane GPA (the range) 2.0 3.8 4.0 x (the domain) f(x) (the range) -7 -6 -5 -4 3.5 -3 2.9 -2 -1 -4 -1 Ordered Pairs: Determining if a mapping represents a function 2 The "ordered pair" approach: 5 A list of (x,y) pairs. Each pair is 'ordered', in the sense that the x value is always first. 8 -3 -2 -1 0 1 2 3 The idea is that x is the input (the domain/starting set), and y is the output (range/ending set). 11 The same intuition applies here, as applies to the mapping: if there is an x value that has two y values, then the collection of pairings is not a function. It may be useful to review this later, as a segue into the "vertical line test" Math 110 Page 1 / 3 Math 110 Chapter 2, Section 2 f ( x) x 4 Page 2 / 3 Function Notation: Important Points: 1. f(x) is a NOTATION It's defined to mean (in this case) x - 4 It does NOT mean multiplication 2. You'll know whether it means multiplication or as a stand-in for a function by the context (i.e., what's around it). In other words, if the problem says "Do something with the following expression, where f(x) = 3x + 2", f(x) means that it's that function, instead of multiplication. 3. Why is this f(x) a function? Notice that for any possible value of x, you get one (and only one) answer back out Try it for x = -3, -2, -1, 0, 1, 2, 3 DOMAIN Sometimes, given a function (like f ( x) x 4 ), you can plug in any real number (negative, positive, fractions, decimals, zero, whatever), and the function will work, and produce real numbers. Sometimes, given a function (like f ( x) x4 ), you can plug in ALMOST any real x number (negative, positive, fractions, decimals, zero, whatever), and the function will work, and produce real numbers. SOME values (such as x= 0) will produce an error, since you can't divide by zero. Sometimes, given a function (like f ( x ) x 4 ), you can plug in ANY real numbers (negative, positive, fractions, decimals, zero, whatever), and the function will work, but it won't always produce real numbers (plug in 2, and you'll get a complex number). So if we want to restrict the range to be only real numbers, we'd have to restrict the domain (in this case, it has to be greater than 4)( x > 4 ) Sometimes, given a function (like A(x) = x2, where A(x) is used to find the area of a square, and x is the length of one side of the square), the domain can be restricted because we know that the physical (and/or geometric) basis of the function eliminates certain values (we can't have a negative length for a side of the box) Operations on Functions: We'll be skipping this for now. Math 110 Page 2 / 3 X Math 110 Chapter 2, Section 2 10 9 8 7 6 5 4 3 2 1 0 -1 0 1 -1 -9 -8 -7 -6 -5 -4 -3 -2 -1 -2 0 -3 -4 -5 -6 -7 -8 -9 -10 Page 3 / 3 2 3 4 5 6 7 8 9 10 Y Math 110 Page 3 / 3