Mathematics - Functions Question with Solution | TestHub

(a) $f$ is not one-one as $f$ cuts the $x$-axis twice in $(0,2\pi ]$. In fact, $f$ is continuous and achieves every real number infinitely many times.

(b) $f$ is monotone as $f^{\prime }(x)=1-{\sin }^{2}x\ge 0$ for all $x$. Also, lim x→−∞ f(x) = −∞ and lim x→+∞ f(x) = +∞. Since $f$ is continuous, it is bijective.

(c) $f(x)=x+|x|=\{\begin{array}{ll}2x& ifx\ge 0\\ 0& ifx\le 0\end{array}$ is not one-one as $f$ is a constant function for $x\le 0$.

(d) $f$ is monotone as $f^{\prime }(x)\le 0$ for all $x$. As in (b), $f$ is bijective.