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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.
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Solution
f is neither one-one nor onto.
Explanation:
f : R → R is defined as f(x) = x4
Let x, y ∈ R such that f(x) = f(y).
⇒ x4 = y4
⇒ x = ±y
∴ f(x1) = f(x2) does not imply that x1 = x2.
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.
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