Properties of Composite Function
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Q.
Let A = {1, 2, 3, 4}, B = {1, 5, 9, 11, 15, 16} and f = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}. Are the following true?
(i) f is a relation from A to B
(ii) f is a function from A to B
Justify your answer in each case.
Q. Let f(x)=sin(π6sin(π2sinx)) for all x∈R and g(x)=π2sinx for all x∈R. Let (f∘g)(x) denotes f(g(x)) and (g∘f)(x) denotes g(f(x)). Then which of the following is (are) true?
- Range of f is [−12, 12]
- Range of f∘g is [−12, 12]
- limx→0f(x)g(x)=π6
- There is an x∈R such that (g∘f)(x)=1
Q. Let f(x)=sin[π6 sin(π2sin x)]∀xϵR and g(x)=π2sinx∀xϵR. Let (fog) (x) denotes f{g(x)} and (gof) (a) denotes g{f(x)}. Then, which of the following is / are true?
- Range of f is [−12, 12]
- Range of fog is [−12, 12]
- limx→0 f(x)g(x)=π6
- There is an x ϵR such that (gof) (x) = 1