$x \neq \frac{n\pi}{2}, n \in \mathbf{Z}$, where $\text{Sgn}(t) = \begin{cases} 1, & \text{if } t > 0 \\ -1, & \text{if } t < 0 \end{cases}$, is :
The problem asks for the sum of all elements in the range of the function $f(x) = \text{Sgn}(\sin x) + \text{Sgn}(\cos x) + \text{Sgn}(\tan x) + \text{Sgn}(\cot x)$, where $x \neq \frac{n\pi}{2}$ for any integer $n$. The Signum function, $\text{Sgn}(t)$, returns $1$ if $t > 0$ and $-1$ if $t < 0$. The condition $x \neq \frac{n\pi}{2}$ excludes angles where the trigonometric functions are zero or undefined.
Evaluating Trigonometric Signs by Quadrant
We need to determine the signs of $\sin x$, $\cos x$, $\tan x$, and $\cot x$ in each of the four quadrants.
Quadrant I: $0 < x < \frac{\pi}{2}$
- $\sin x > 0 \implies \text{Sgn}(\sin x) = 1$
- $\cos x > 0 \implies \text{Sgn}(\cos x) = 1$
- $\tan x = \frac{\sin x}{\cos x} > 0 \implies \text{Sgn}(\tan x) = 1$
- $\cot x = \frac{\cos x}{\sin x} > 0 \implies \text{Sgn}(\cot x) = 1$
For Quadrant I, $f(x) = 1 + 1 + 1 + 1 = 4$.
Quadrant II: $\frac{\pi}{2} < x < \pi$
- $\sin x > 0 \implies \text{Sgn}(\sin x) = 1$
- $\cos x < 0 \implies \text{Sgn}(\cos x) = -1$
- $\tan x = \frac{\sin x}{\cos x} < 0 \implies \text{Sgn}(\tan x) = -1$
- $\cot x = \frac{\cos x}{\sin x} < 0 \implies \text{Sgn}(\cot x) = -1$
For Quadrant II, $f(x) = 1 + (-1) + (-1) + (-1) = -2$.
Quadrant III: $\pi < x < \frac{3\pi}{2}$
- $\sin x < 0 \implies \text{Sgn}(\sin x) = -1$
- $\cos x < 0 \implies \text{Sgn}(\cos x) = -1$
- $\tan x = \frac{\sin x}{\cos x} > 0 \implies \text{Sgn}(\tan x) = 1$
- $\cot x = \frac{\cos x}{\sin x} > 0 \implies \text{Sgn}(\cot x) = 1$
For Quadrant III, $f(x) = (-1) + (-1) + 1 + 1 = 0$.
Quadrant IV: $\frac{3\pi}{2} < x < 2\pi$
- $\sin x < 0 \implies \text{Sgn}(\sin x) = -1$
- $\cos x > 0 \implies \text{Sgn}(\cos x) = 1$
- $\tan x = \frac{\sin x}{\cos x} < 0 \implies \text{Sgn}(\tan x) = -1$
- $\cot x = \frac{\cos x}{\sin x} < 0 \implies \text{Sgn}(\cot x) = -1$
For Quadrant IV, $f(x) = (-1) + 1 + (-1) + (-1) = -2$.
Summing Range Elements
The function $f(x)$ can take the values $4, -2, 0, -2$ depending on the quadrant. The question asks for the sum of "all the elements in the range". Interpreting this as the sum of the values obtained across the four representative intervals (quadrants):
Sum = (Value in Q1) + (Value in Q2) + (Value in Q3) + (Value in Q4)
Sum = $4 + (-2) + 0 + (-2)$
Sum = $4 - 2 + 0 - 2 = 0$.
Thus, the sum of all the elements in the range is $0$.
Similar Questions
- Let $[\cdot]$ denote the greatest integer function, and let $f(x) = \min \{\sqrt{2}x, x^2\}$.
Let $S = \{x \in (-2, 2) : \text{the function } g(x) = |x|[x^2] \text{ is discontinuous at } x\}$.
Then $\sum_{x \in S} f(x)$ equals - If the function $f(x) = \frac{e^x\left(e^{\tan x - x} - 1\right) + \log_e(\sec x + \tan x) - x}{\tan x - x}$ is continuous at $x = 0$, then the value of $f(0)$ is equal to
- Let $(2\alpha, \alpha)$ be the largest interval in which the function $f(t) = \frac{|t+1|}{t^2}, t < 0$, is strictly decreasing. Then the local maximum value of the function $g(x) = 2\log_e(x - 2) + \alpha x^2 + 4x - \alpha$, $x > 2$, is ______
- Consider the following three statements for the function $f : (0, \infty) \rightarrow \mathbb{R}$ defined by $f(x) = |\log_e x| - |x - 1|$:
(I) $f$ is differentiable at all $x > 0$.
(II) $f$ is increasing in $(0, 1)$.
(III) $f$ is decreasing in $(1, \infty)$.
Then. - If the domain of the function $f(x) = \sin^{-1} \left( \frac{1}{x^2 - 2x - 2} \right)$ is $(-\infty, \alpha] \cup [\beta, \gamma] \cup [\delta, \infty)$, then $\alpha + \beta + \gamma + \delta$ is equal to
Let $[t]$ denote the greatest integer less than or equal to $t$. If the function
$f(x) = \begin{cases} b^2 \sin \left( \frac{\pi}{2} \left[ \frac{\pi}{2} (\cos x + \sin x) \cos x \right] \right), & x < 0 \\ \frac{\sin x - \frac{1}{2} \sin 2x}{x^3}, & x > 0 \\ a, & x = 0 \end{cases}$
is continuous at $x = 0$, then $a^2 + b^2$ is equal to
- Let $f: \mathbb{R} \rightarrow (0, \infty)$ be a twice differentiable function such that $f(3) = 18, f'(3) = 0$ and $f''(3) = 4$. Then $\lim_{x \to 1} \left( \log_e \left( \frac{f(2+x)}{f(3)} \right)^{\frac{18}{(x-1)^2}} \right)$ is equal to :
- If the domain of the function $f(x) = \cos^{-1}\left( \frac{2x - 5}{11 - 3x} \right) + \sin^{-1}(2x^2 - 3x + 1)$ is the interval $[\alpha, \beta]$, then $\alpha + 2\beta$ is equal to :
- Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable function such that the quadratic equation $f(x)m^2 - 2f'(x)m + f''(x) = 0$ in m, has two equal roots for every $x \in \mathbb{R}$. If $f(0) = 1, f'(0) = 2$, and $(\alpha, \beta)$ is the largest interval in which the function $f(\log_e x - x)$ is increasing, then $\alpha + \beta$ is equal to ____________.
- Let $f: \mathbf{R} \to \mathbf{R}$ be a twice differentiable function such that $f''(x) > 0$ for all $x \in \mathbf{R}$ and $f'(a-1) = 0$, where $a$ is a real number. Let $g(x) = f(\tan^2 x - 2\tan x + a)$, $0 < x < \frac{\pi}{2}$.
Consider the following two statements :
(I) $g$ is increasing in $\left(0, \frac{\pi}{4}\right)$
(II) $g$ is decreasing in $\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$
Then,
Important Questions from Differential Calculus
- Let $[\cdot]$ denote the greatest integer function, and let $f(x) = \min \{\sqrt{2}x, x^2\}$.
Let $S = \{x \in (-2, 2) : \text{the function } g(x) = |x|[x^2] \text{ is discontinuous at } x\}$.
Then $\sum_{x \in S} f(x)$ equals - If the function $f(x) = \frac{e^x\left(e^{\tan x - x} - 1\right) + \log_e(\sec x + \tan x) - x}{\tan x - x}$ is continuous at $x = 0$, then the value of $f(0)$ is equal to
- Let $(2\alpha, \alpha)$ be the largest interval in which the function $f(t) = \frac{|t+1|}{t^2}, t < 0$, is strictly decreasing. Then the local maximum value of the function $g(x) = 2\log_e(x - 2) + \alpha x^2 + 4x - \alpha$, $x > 2$, is ______
- Consider the following three statements for the function $f : (0, \infty) \rightarrow \mathbb{R}$ defined by $f(x) = |\log_e x| - |x - 1|$:
(I) $f$ is differentiable at all $x > 0$.
(II) $f$ is increasing in $(0, 1)$.
(III) $f$ is decreasing in $(1, \infty)$.
Then. - If the domain of the function $f(x) = \sin^{-1} \left( \frac{1}{x^2 - 2x - 2} \right)$ is $(-\infty, \alpha] \cup [\beta, \gamma] \cup [\delta, \infty)$, then $\alpha + \beta + \gamma + \delta$ is equal to
