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Question
Suppose f1 and f2 are non-zero one-one functions from R to R. Is `f_1 / f^2` necessarily one - one? Justify your answer. Here,`f_1/f_2 : R → R is given by (f_1/f_2) (x) = (f_1(x))/(f_2 (x)) for all x in R .`
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Solution
We know that f1: R → R, given by f1(x)=x3 and f2(x)=x are one-one.
Injectivity of f1:
Let x and y be two elements in the domain R, such that
f1 (x) = f2 (y) ⇒ x3= y ⇒ x = `3 sqrty in R`
So, f1 is one-one.
Injectivity of f2:
Let x and y be two elements in the domain R, such that
f2(x) = f2 (y) ⇒ x= y ⇒ x∈R.
So, f2 is one-one
proving `f_1 / f_2` is not one - one :
Given that `f_1/f_2 (x) = f_1(x)/f_2(x) = x^2/x = x^2`
Let x and y be two elements in the domain R, such that
`f_1/(f_2) (x) = f_1/(f_2) (y)`
⇒ x2= y2
⇒ x = ±y
So, `f_1/f_2` is not one - one
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