Question

Which of the following functions are onto functions?

• A. $f:R\to [-1,1]$ as $f\left(x\right)=sin\left(x\right)$
• B. $f:R\to [-1,1]$ as $f\left(x\right)=cos\left(x\right)$
• C. $f:R\to (0,\infty )$ as $f\left(x\right)=e^x$
• D. $f:R\to (-\infty ,\infty )$ as $f\left(x\right)=x^3$

Answer 1

Answer: (A), (B), (C), (D)

$f:R\to (-\infty ,\infty )$ as $f\left(x\right)=x^3$

A function $f:A\to B$ is onto if every element of the co-domain has a pre-image.
i.e., $f$ is onto if range equals co-domain.

We know that the range of the functions $\text{sin}x$ and $\text{cos}x$ is [-1,1]. Since range equals co-domain, both of these functions are onto.
Similarly, we can check for the other three functions and we’ll find that all functions are onto.