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Choose the correct option.
Statement I: f: mathbbR → mathbbR given by f(x) = 1/(x-2), is neither injective nor surjective…
Statement I: $f: \mathbb{R} \to \mathbb{R}$ given by $f(x) = \frac{1}{x-2}$, is neither injective nor surjective.
Statement II: $f: \mathbb{Z} \to \mathbb{Z}$ given by $f(x) = \sqrt[3]{x^9}$, is neither injective nor surjective.
- (a)Both the statements are true.
- (b)Both the statements are false.
- (c)Statement I is true, and Statement II is false.
- (d)Statement I is false, and Statement II is true.
Answer
Answer
Correct option: (b)
Answer: (b) Both the statements are false.
I: $f(x) = \frac{1}{x-2}$ is one-one (if $\frac{1}{x_1-2} = \frac{1}{x_2-2}$ then $x_1 = x_2$) but never takes the value 0, so it is not onto: it is not 'neither', hence the statement is false.
II: $f(x) = \sqrt[3]{x^9} = x^3$ is one-one on $\mathbb{Z}$ (injective), though not onto (2 has no integer cube root), so 'neither injective nor surjective' is false.