Which of these functions from R to R is onto?

Question 112

MediumOriginal question

Which of these functions from $\mathbb{R}$ to $\mathbb{R}$ is onto?

f(x) = x^{2} - 3x

Correct answerYour answer

f(x) = \frac{1}{1 + x^{2}}

Correct answerYour answer

f(x) = x^{3} - 3x

Correct answerYour answer

f(x) = 3^{x}

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer C

Correct answer: C

For any real $y$ , $x^{3} - 3x - y = 0$ is a cubic equation, and a cubic always has a real root: onto. (It is not one-one, as $f(0) = f(\sqrt{3}) = 0$ , but onto does not need that.)
The others miss values: $x^{2} - 3x \ge -\frac{9}{4}$ , $0 < \frac{1}{1 + x^{2}} \le 1$ and $3^{x} > 0$ .

Worth noting

A quadratic has a lowest value: $x^{2} - 3x = \left(x - \frac{3}{2}\right)^{2} - \frac{9}{4}$ never goes below $-\frac{9}{4}$ .

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