Question

Which of the following is an onto function

• A. $f:[0,\pi ]\to [-11],f\left(x\right)=\sin x$
• B. $f:[0,\pi ]\to [-1,1],f\left(x\right)=\cos x$
• C. $f:R\to R,f\left(x\right)=e^x$
• D. $f:Q\to R,f\left(x\right)=x^3$

Answer 1

The correct option is B $f:[0,\pi ]\to [-1,1],f\left(x\right)=\cos x$

A function is called onto iff, Range$=$ co-domain.
A. In $[0,\pi ]$, Range of $\sin x$ is $[0,1]\Rightarrow$ Not onto.
B. In $[0,\pi ]$, Range of $\cos x$ is $[-1,1]=$co-domain $\Rightarrow$onto function.
C. Range of $e^x$ is $(0,\infty )\ne$ co-domain.
D. Range of $x^3$ is not R, since $x\in Q$