Questions {a, {1, = = = f b, c} f (a) f (b) f (c) f Let be the function from to 2, 3} such that 2, 3, and 1. Is invertible, and if it is, what is its inverse? : Z → Z = + f f (x) x f Let be such that 1. Is invertible, and if it is, what is its inverse? Let f be the function from R to R with f (x) = x2. Is f invertible, and if it is, what is its inverse? Composition of functions Suppose f: AB and g: BC. f(a)=b and g(b)=c for aA, bB, and cC. Then the composition h = g f is a function h: AC or (g f): AC Note: g f is read as “g composed with f”. h(a)=_________________________ Note that g ◦ f cannot be defined unless the range of f is a subset of the domain of g Function Review Questions Let f : AB and g: BC. If f and g are one-to-one, is g ◦ f also one-to-one? If f and g are onto, is g ◦ f also onto? If g and g ◦ f are onto, is f also onto? N N Give an example of a function from to that is one-to-one but not onto. onto but not one-to-one. both onto and one-to-one (but different from the identity function). neither one-to-one nor onto. Start on Sequences and Summations (2.4) A sequence is a function from a subset of the set of integers (usually either the set {0, 1, 2, . . .} or the set {1, 2, 3, . . .}) to a set S. We use the notation an to denote the image of the integer n. We call an a term of the sequence. Consider the sequence where an=2n. In that case, the sequence, starting with a1 is 2, 4, 6, 8, etc. Consider the sequence where an=. In that case, what is the sequence?
Questions Let f be the function from {a, b, c} to {1, 2, 3} such that f (a) = 2, f (b) = 3, and f (c) = 1. Is f invertible, and if it is, what is its inverse? Let f : Z → Z be such that f (x) = x + 1. Is f invertible, and if it is, what is its inverse? Let f be the function from R to R with f (x) = x2. Is f invertible, and if it is, what is its inverse? Composition of functions Suppose f: AB and g: BC. f(a)=b and g(b)=c for aA, bB, and cC. Then the composition h = g f is a function h: AC or (g f): AC Note: g f is read as “g composed with f”. h(a)=_________________________ Note that g ◦ f cannot be defined unless the range of f is a subset of the domain of g Function Review Questions Let f : AB and g: BC. If f and g are one-to-one, is g ◦ f also one-to-one? If f and g are onto, is g ◦ f also onto? If g and g ◦ f are onto, is f also onto? Give an example of a function from N to N that is one-to-one but not onto. onto but not one-to-one. both onto and one-to-one (but different from the identity function). neither one-to-one nor onto. Start on Sequences and Summations (2.4) A sequence is a function from a subset of the set of integers (usually either the set {0, 1, 2, . . .} or the set {1, 2, 3, . . .}) to a set S. We use the notation an to denote the image of the integer n. We call an a term of the sequence. Consider the sequence where an=2n. In that case, the sequence, starting with a1 is 2, 4, 6, 8, etc. Consider the sequence where an=. In that case, what is the sequence?
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