Question

Which of the following statements is (are) CORRECT?

• A. The fundamental period of the function $f\left(x\right)=\text{cosec}\left(\frac{x}{2}\right)+\sec \left(\frac{x}{3}\right)+\cot \left(\frac{x}{4}\right)+\tan \left(\frac{x}{5}\right)+\cos \left(\frac{x}{6}\right)+\sin \left(\frac{x}{7}\right)$ is $420\pi$.
• B. If $f$ is a function such that $f\left(s\right)=f\left(t\right),$ then $s=t$.
• C. A vertical line intersects the graph of a function at most once.
• D. If $h\left(x\right)=\{\begin{array}{cc}{x}^2,& x\le 0\\ x,& x>0\end{array},$ then $h\left(h\right(x\left)\right)=h\left(x\right)\forall x\in R$.

Answer 1

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

The correct options are
A The fundamental period of the function $f\left(x\right)=\text{cosec}\left(\frac{x}{2}\right)+\sec \left(\frac{x}{3}\right)+\cot \left(\frac{x}{4}\right)+\tan \left(\frac{x}{5}\right)+\cos \left(\frac{x}{6}\right)+\sin \left(\frac{x}{7}\right)$ is $420\pi$.
C A vertical line intersects the graph of a function at most once.
D If $h\left(x\right)=\{\begin{array}{cc}{x}^2,& x\le 0\\ x,& x>0\end{array},$ then $h\left(h\right(x\left)\right)=h\left(x\right)\forall x\in R$.

Period of $f\left(x\right)=$ L.C.M. of $\{\frac{2\pi }{\left(\frac{1}{2}\right)},\frac{2\pi }{\left(\frac{1}{3}\right)},\frac{\pi }{\left(\frac{1}{4}\right)},\frac{\pi }{\left(\frac{1}{5}\right)},\frac{2\pi }{\left(\frac{1}{6}\right)},\frac{2\pi }{\left(\frac{1}{7}\right)}\}$

$=$ L.C.M. of $\{4\pi ,6\pi ,4\pi ,5\pi ,12\pi ,14\pi \}$
$=420\pi$

Clearly, $f\left(s\right)=f\left(t\right)$ does not imply $s=t$
If $f$ is one-one and $f\left(s\right)=f\left(t\right)$, then $s=t$

Obviously, any vertical line intersects the graph of a function at most once.

We have $h\left(x\right)=\{\begin{array}{cc}{x}^2,& x\le 0\\ x,& x>0\end{array}$
But $h\left(x\right)\ge 0\forall x\in R$
So, $h\left(h\right(x\left)\right)=\{\begin{array}{cc}{x}^2,& x\le 0\\ x,& x>0\end{array}=h\left(x\right),\forall x\in R$