CSC165H1 Lecture 6: lecture 6
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CSC165H1 Full Course Notes
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Definition: a function from a set a to a set b assigns to each a a , a unique element f(x) A is called thd domain of f. B is called the target(also range") of f. Notation: f:a b means that f is a function from a b. F(a) is the image of f, it is the set of { f(a)| a a}. Example3: let h: z x z z be h ((m,n))=m^2 + n^2. Definition: the graph of a function f: a b is the set {(a,f(a))| a a}. For example: let f: r r be given by f(x)=x/(1+x^2), show that f(r)=[- , ]. Rough work: - x/(1+x^2) . The last inequality is clearly true, we conclude that - x/(1+x^2) . Now let"s prove that [- , ] we need to show that y=f(x)=x/(1+x^2) for some x r. This is a quadratic with a=y, b=-1, c=y, so b^2-4ac=(-1)^2-4y^2=1-4y^2. Since - y , we get y^2 .
