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Question
Let f: R → R be defined as f(x) = x4. Choose the correct answer.
Options
f is one - one onto
f is many - one onto
f is one - one but not onto
f is neither one - one nor onto
Solution
f is neither one - one nor onto
Explanation:
f: R → R is defined as `f(x) = x^4`
Let x, y ∈ R such that f(x) = f(y).
` =>x^4 = y^4`
`=> x=+-y`
∴`f(x_1) = f(x_2)` does not imply that `x_1 = x_2`
For instance,
f(1) = f(-1) = 1
∴ f is not one-one.
Consider an element 2 in co-domain R. It is clear that there does not exist any x in domain R such that f(x) = 2.
∴ f is not onto.
Hence, function f is neither one-one nor onto.
The correct answer is D.
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