Question:
The range of the function $f \left(x\right)=\frac{x}{1+\left|x\right|}, x\,\in\,R,$ is
The range of the function $f \left(x\right)=\frac{x}{1+\left|x\right|}, x\,\in\,R,$ is
Updated On: Aug 15, 2022
- $R$
- $(-1, 1)$
- $R-\left\{0\right\}$
- $[-1, 1]$
Hide Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
$f \left(x\right)=\frac{x}{1+\left|x\right|}, x\,\in\,R$
If $x>0, \left|x\right|=x \Rightarrow f \left(x\right)=\frac{.x}{1+x}$
which is not defined for $x = -1$
If $x>0, \left|x\right|=x \Rightarrow f \left(x\right)=\frac{.x}{1+x}$
which is not defined for $x = 1$
Thus $f (x)$ defined for all values of $R$ except 1 and-1
Hence, range $= (-1, 1).$
Was this answer helpful?
0
0
Top Questions on Application of derivatives
- Let \(A\) be the region enclosed by the parabola \(y^2 = 2x\) and the line \(x = 24\). Then the maximum area of the rectangle inscribed in the region \(A\) is ________.
- JEE Main - 2024
- Mathematics
- Application of derivatives
- For the function \(f(x) = \sin x + 3x - \frac{2}{\pi}(x^2 + x), \quad x \in \left[0, \frac{\pi}{2}\right],\)consider the following two statements:
1. \( f \) is increasing in \( \left(0, \frac{\pi}{2}\right) \).
2. \( f' \) is decreasing in \( \left(0, \frac{\pi}{2}\right) \).
Between the above two statements- JEE Main - 2024
- Mathematics
- Application of derivatives
- Let the set of all values of \( p \), for which \[ f(x) = (p^2 - 6p + 8)(\sin^2 2x - \cos^2 2x) + 2(2 - p)x + 7 \] does not have any critical point, be the interval \( (a, b) \). Then \( 16ab \) is equal to _____ .
- JEE Main - 2024
- Mathematics
- Application of derivatives
Area of region enclosed by curve y=x3 and its tangent at (–1,–1)
- JEE Main - 2023
- Mathematics
- Application of derivatives
The minimum of \(f(x)=\sqrt{(10-x^2)}\) in the interval \([-3,2]\) is
- KEAM - 2023
- Mathematics
- Application of derivatives
View More Questions
Questions Asked in AIEEE exam
- This question has Statement 1 and Statement 2. Of the four choices given after the Statements, choose the one that best describes the two Statements. It is not possible to make a sphere of capacity $1$ farad using a conducting material. It is possible for earth as its radius is $6.4 \times 10^6\,m$.
- AIEEE - 2012
- electrostatic potential and capacitance
- A steel wire can sustain $100\, kg$ weight without breaking. If the wire is cut into two equal parts, each part can sustain a weight of
- AIEEE - 2012
- mechanical properties of solids
- The limiting line in Balmer series will have a frequency of (Rydberg constant, $R_{\infty}=3.29\times10^{15}$ cycles/s)
- AIEEE - 2012
- Electromagnetic Spectrum
- Which of the plots shown in the figure represents speed (v) of the electron in a hydrogen atom as a function of the principal quantum number (n)?
- The ratio of number of oxygen atoms (O) in 16.0 g ozone $(O_3), \,28.0\, g$ carbon monoxide $(CO)$ and $16.0$ oxygen $(O_2)$ is (Atomic mass: $C = 12,0 = 16$ and Avogadro?? constant $N_A = 6.0 x 10^{23}\, mol^{-1}$)
- AIEEE - 2012
- Mole concept and Molar Masses
View More Questions
Concepts Used:
Application of Derivatives
Various Applications of Derivatives-
Rate of Change of Quantities:
If some other quantity ‘y’ causes some change in a quantity of surely ‘x’, in view of the fact that an equation of the form y = f(x) gets consistently pleased, i.e, ‘y’ is a function of ‘x’ then the rate of change of ‘y’ related to ‘x’ is to be given by
\(\frac{\triangle y}{\triangle x}=\frac{y_2-y_1}{x_2-x_1}\)
This is also known to be as the Average Rate of Change.
Increasing and Decreasing Function:
Consider y = f(x) be a differentiable function (whose derivative exists at all points in the domain) in an interval x = (a,b).
- If for any two points x1 and x2 in the interval x such a manner that x1 < x2, there holds an inequality f(x1) ≤ f(x2); then the function f(x) is known as increasing in this interval.
- Likewise, if for any two points x1 and x2 in the interval x such a manner that x1 < x2, there holds an inequality f(x1) ≥ f(x2); then the function f(x) is known as decreasing in this interval.
- The functions are commonly known as strictly increasing or decreasing functions, given the inequalities are strict: f(x1) < f(x2) for strictly increasing and f(x1) > f(x2) for strictly decreasing.
Read More: Application of Derivatives