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1. Let f:(0,1)→R be defined by f(x)=3arcsin(x) for all x∈dom(f). Let g:[−2π,2π]→R be any function with this domain. De
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1. Let f:(0,1)→R be defined by f(x)=3arcsin(x) for all x∈dom(f). Let g:[−2π,2π]→R be any function with this domain. De
1. Let f:(0,1)→R be defined by f(x)=3arcsin(x) for all x∈dom(f). Let g:[−2π,2π]→R be any function with this domain. Define the composite function h=g o f on the maximal domain given by these definitions. Finally, define p:dom(h)→R by p(x)=h(x)/x for all x∈ dom (h). (a) Determine dom(h). (Note: Do not assign an expression for g(x) ).) (b) Now suppose that g(x)=sin(x) for all x∈dom(g). Using only trigonometric identities, determine an algebraic expression for g(3x) in terms of g(x) only. (c) Determine an algebrajc expression for h(x). (d) Justify that p has an inverse function p−1 by arguing that p is one-to-one. (e) Determine the domain and range of p−1. (f) Determine an algebraic expression for p−1(x).
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