Question

Which of the following functions defined from $(-\infty ,\infty )$ to $(-\infty ,\infty )$ is invertible ?

• A. $f\left(x\right)=\sin (2x+3)$
• B. $f\left(x\right)=x^2+4$
• C. $f\left(x\right)=x^3$
• D. $f\left(x\right)=\cos x$

Answer 1

Answer: (C)

$f\left(x\right)=x^3$

$f\left(x\right)=\sin (2x+3)$ lies only from $[-1,1]$ and hence is not onto.
So, $f\left(x\right)$ is not bijective
$\therefore$ it is not invertible
$f\left(x\right)=x^2+4$ is always positive and so not onto.
So, $f\left(x\right)$ is not bijective
$\therefore$ It is also not invertible.
$f\left(x\right)=x^3$ varies from $(-\infty ,\infty )$ as $x$ varies from $(-\infty ,\infty )$
So, $f\left(x\right)$ is bijective
$\therefore$ It is invertible.
$f\left(x\right)=\cos x$ always lies between $[-1,1]$ and so is into function.
So, $f\left(x\right)$ is not bijective
Hence, not invertible.