Which of the following statements are true? I. f : Z → Z, f(n) = n^2, is one-one. II. f : N → N, f(n) = n^2 + n, is one-one. III. f : R → R, f(x) = x^3 - x, is one-one. IV. f : Z → Z, f(n) = 3n - 7, is one-one.

Question 94

MediumOriginal question

Which of the following statements are true?
I. $f : \mathbb{Z} \to \mathbb{Z}$ , $f(n) = n^{2}$ , is one-one.
II. $f : \mathbb{N} \to \mathbb{N}$ , $f(n) = n^{2} + n$ , is one-one.
III. $f : \mathbb{R} \to \mathbb{R}$ , $f(x) = x^{3} - x$ , is one-one.
IV. $f : \mathbb{Z} \to \mathbb{Z}$ , $f(n) = 3n - 7$ , is one-one.

II and IV onlyCorrect answerYour answerI, II and IV onlyCorrect answerYour answerII, III and IV onlyCorrect answerYour answerIV onlyCorrect answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

I is false: $f(-2) = f(2) = 4$ . III is false: $f(-1) = f(0) = f(1) = 0$ .
II is true: $n^{2} + n$ grows as $n$ grows on $\mathbb{N}$ . IV is true: $3a - 7 = 3b - 7$ gives $a = b$ .

Worth noting

$n^{2}$ is one-one on $\mathbb{N}$ , but $\mathbb{Z}$ brings in negatives, and $-2$ and 2 have the same square.

Concept: One-one on different domains
Typical timeTypical: 120 s
Your timeYours: —

The rule this testsOne-one functions

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