7. Given a function f:A B and a set X A, we can construct a new function g:X B b

Question

7. Given a function f:A B and a set X A, we can construct a new function g:X B by defining g (x) = f(x). This process is called "restricting the domain." For instance, if R R is given by f(x) = x2 and we restrict the domain by defining g: [0,00) R by the same formula, g(x) = x2, the result is a one-to-one function. Suppose h: R R is defined by h(x) sin x. Determine a set X so that when we restrict the domain to the new function k:X R defined by k (x) = sin x is one-to-one and has the same range (the range for each of these is the closed interval -1,1].)

Explanation / Answer

From the graph of a one to one function we see that any horizontal line intersects it in at most one point

From the graph of sine function we see that such an interval is between consecutive extrema ie one peak where it takes value 1 and then next time it takes value equal to -1

So such an interval is

X=[-PI/2,PI/2]

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