Question

Which of the following functions are into functions if they are all defined from $R$ to $R$?

• A. $f\left(x\right)=\text{sin}x$
• B. $f\left(x\right)=\text{cos}x$
• C. $f\left(x\right)=e^x$
• D. $f\left(x\right)=x^3$

Answer 1

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

$f\left(x\right)=e^x$

If the range of a function is a proper subset of the co-domain, then the function is into.

Consider $f\left(x\right)=\text{sin}x.$
We know that the range of $\text{sin}x$ is $[-1,1].$
Thus, range $\subset$ co-domain.
i.e., There are elements in the co-domain that have no pre-images. Thus, the function is into.

Consider $f\left(x\right)=\text{cos}x.$
We know that the range of $\text{cos}x$ is $[-1,1].$ Since range $\subset$ co-domain, the function is into.

Consider $f\left(x\right)=e^x.$ This function ranges from $(0,\infty )$ and thus, range $\subset$ co-domain. Therefore, the function is into.

Lastly, $f\left(x\right)=x^3.$ has $R$ as its range and co-domain. Therefore, it is not an into function.