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Unformatted text preview: Son of Functions Margaret M. Fleck 22 February 2010 This lecture finishes covering the main concepts involving functions, though we’ll see more example proofs on Friday. 1 Announcements Remember to bring your ID to Wednesday’s exam. Also notice that the exams from previous term were held in class, so expect our exam to be somewhat longer (but nowhere near twice as long). Office hours Th/Fri this week are cancelled, because there’s no homework due. Bring any last-minute questions to lectures Wednesday, or come to Wednesday office hours (regular and extra) 11-12 (Viraj and Samer) and 3:30 (approx) to 5 (Chen and Adair). 2 Recap Last lecture, we saw that functions are defined with a specific declared do- main (set of input values) and co-domain (set of legit output values). The image of a function is the set of actual values produced if you run all the input values through the function. So the image is a subset of the co-domain, but the two might not be equal. A function f : A → B is onto (surjective) if ∀ y ∈ B, ∃ x ∈ A, f ( x ) = y . That is, the image is all of the co-domain. For example, the function f : R → R defined by f ( x ) = x + 2 is onto. But the function g : R → R g ( x ) = x 2 is not onto, because it never produces negative outputs. g would be onto if we had defined its co-domain to be only the non-negative reals. 1 In this class, most examples of non-onto functions look like cases where you could have defined them to be onto, but the author just didn’t feel like setting up the co-domain precisely. Or, when it comes to computer programs, perhaps the compiler declarations don’t allow you to be sufficiently precise, e.g. there’s no distinct type for non-negative floating point numbers. In some applications, however, it’s critical that certain functions not be onto. For example, in graphics or certain engineering applications, we may wish to map out or draw a curve in 2D space. The whole point of the curve is that it occupies only part of 2D and it is surrounded by whitespace. These curves are often specified “parametrically,” using functions that map into, but not onto, 2D. For example, we can specify a (unit) circle as the image of a function f : [0 , 1] → R 2 defined by f ( x ) = (cos 2 πx, sin 2 πx ). If you think of the input values as time, then f shows the track of a pen or robot as it goes around the circle. The cosine and sine are the x and y coordinates of its position....
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- Spring '08
- Inverse function, codomain, input values, output value