Question

The set of elements in the range of the function
f(x)=syn(sin x)- sgn(cosx) - sgn(tanx) - sgn(cotx),
(x is taken from the domain of the function) has p
distinct elements then p is equal to
Answer:
oo 01 02 03 04 05 06 07
08 Og​

Answer 1

Let n ∈ Z.

The domain of f(x) = sgn(sin x) - sgn(cos x) - sgn(tan x) - sgn(cot x) is,

$\small\text{ \longrightarrow x\in\mathbb{R}-\left\{2n\pi,\ 2n\pi+\dfrac{\pi}{2},\ 2n\pi+\pi,\ 2n\pi+\dfrac{3\pi}{2}\right\} }$

because tan x is not defined for $\small\text{ x\in\left\{2n\pi+\dfrac{\pi}{2},\ 2n\pi+\dfrac{3\pi}{2}\right\} }$ and cot x is not defined for $\smallx\in\left\{2n\pi,\ 2n\pi+\pi\right\}.$

Signum function is defined as,

$\small\text{ \longrightarrow\mathrm{sgn}(x)=\left\{\begin{array}{lr}-1,&x<0\\\\0,&x=0\\\\1,&x>0\end{array}\right. }$

Thus the following are defined.

$\small\text{ \longrightarrow\mathrm{sgn}(\sin x)=\left\{\begin{array}{ll}-1,&x\in\left(2n\pi+\pi,\ 2n\pi+2\pi\right)\\\\0,&x\in\left\{2n\pi,\ 2n\pi+\pi\right\}\\\\1,&x\in\left(2n\pi,\ 2n\pi+\pi\right)\end{array}\right. }$

$\small\text{ \longrightarrow\mathrm{sgn}(\cos x)=\left\{\begin{array}{ll}-1,&x\in\left(2n\pi+\dfrac{\pi}{2},\ 2n\pi+\dfrac{3\pi}{2}\right)\\\\0,&x\in\left\{2n\pi+\dfrac{\pi}{2},\ 2n\pi+\dfrac{3\pi}{2}\right\}\\\\1,&x\in\left(2n\pi,\ 2n\pi+\dfrac{\pi}{2}\right)\cup\left(2n\pi+\dfrac{3\pi}{2},\ 2n\pi+2\pi\right)\end{array}\right. }$

$\small\text{ \longrightarrow\mathrm{sgn}(\tan x)=\mathrm{sgn}(\cot x)=\left\{\begin{array}{ll}-1,&x\in\left(2n\pi+\dfrac{\pi}{2},\ 2n\pi+\pi\right)\cup\left(2n\pi+\dfrac{3\pi}{2},\ 2n\pi+2\pi\right)\\\\0,&x\in\left\{2n\pi,\ 2n\pi+\pi\right\}\\\\1,&x\in\left(2n\pi,\ 2n\pi+\dfrac{\pi}{2}\right)\cup\left(2n\pi+\pi,\ 2n\pi+\dfrac{3\pi}{2}\right)\end{array}\right. }$We consider each possible interval common to all the interval in the cases (one case taken from one function at a time), then identify values of each signum function in those intervals, then evaluate the value of f(x) in them.

Then we get the following definition for f(x),

$\small\text{ \longrightarrow f(x)=\left\{\begin{array}{ll}1-1-1-1,&x\in\left(2n\pi,\ 2n\pi+\dfrac{\pi}{2}\right)\\\\1-(-1)-(-1)-(-1),&x\in\left(2n\pi+\dfrac{\pi}{2},\ 2n\pi+\pi\right)\\\\(-1)-(-1)-1-1,&x\in\left(2n\pi+\pi,\ 2n\pi+\dfrac{3\pi}{2}\right)\\\\(-1)-1-(-1)-(-1),&x\in\left(2n\pi+\dfrac{3\pi}{2},\ 2n\pi+2\pi\right)\end{array}\right. }$

$\small\text{ \longrightarrow f(x)=\left\{\begin{array}{ll}-2,&x\in\left(2n\pi,\ 2n\pi+\dfrac{\pi}{2}\right)\\\\4,&x\in\left(2n\pi+\dfrac{\pi}{2},\ 2n\pi+\pi\right)\\\\-2,&x\in\left(2n\pi+\pi,\ 2n\pi+\dfrac{3\pi}{2}\right)\\\\0,&x\in\left(2n\pi+\dfrac{3\pi}{2},\ 2n\pi+2\pi\right)\end{array}\right. }$

or,

$\small\text{ \longrightarrow f(x)=\left\{\begin{array}{ll}-2,&x\in\left(2n\pi,\ 2n\pi+\dfrac{\pi}{2}\right)\cup\left(2n\pi+\pi,\ 2n\pi+\dfrac{3\pi}{2}\right)\\\\0,&x\in\left(2n\pi+\dfrac{3\pi}{2},\ 2n\pi+2\pi\right)\\\\4,&x\in\left(2n\pi+\dfrac{\pi}{2},\ 2n\pi+\pi\right)\end{array}\right. }$

Here's other way to get this definition.

• For $\smallx\in\left(0,\ \dfrac{\pi}{2}\right)$ all functions are positive so each signum function equals 1.

• For $\smallx\in\left(\dfrac{\pi}{2},\ \pi\right)$ all functions except sin x are negative so sgn(sin x) = 1 and others equal -1.

• For $\smallx\in\left(\pi,\ \dfrac{3\pi}{2}\right)$ all functions except tan x and cot x are negative so sgn(tan x) = sgn(cot x) = 1 and others equal -1.

• For $\smallx\in\left(\dfrac{3\pi}{2},\ 2\pi\right)$ all functions except cos x are negative so sgn(cos x) = 1 and others equal -1.

• As we already found, f(x) is not defined for $\small\text{ x\in\left\{2n\pi,\ 2n\pi+\dfrac{\pi}{2},\ 2n\pi+\pi,\ 2n\pi+\dfrac{3\pi}{2}\right\}. }$

Well the range of f(x) is,

$\small\longrightarrow f(x)\in\{-2,\ 0,\ 4\}$

There are 3 distinct elements in the range. Hence p = 3.