Question:
The domain of the real valued function
\[
f(x)=\frac{\log_2(x+3)}{\sqrt{x^2+3x+2}}
\]
is
The domain of the real valued function
\[
f(x)=\frac{\log_2(x+3)}{\sqrt{x^2+3x+2}}
\]
is
Show Hint
For functions involving logarithms and square roots in denominator:
\[
\text{(i) logarithm argument } \gt 0
\]
and
\[
\text{(ii) quantity under square root in denominator } \gt 0
\]
must both hold simultaneously.
Updated On: Jun 26, 2026
- \((-3,\infty)\)
- \((-3,-1)\cup(-1,\infty)\)
- \((-3,-2)\cup(-2,-1)\cup(-1,\infty)\)
- \((-3,-2)\cup(-1,\infty)\)
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Step 1: Condition for logarithm to exist.
For \[ \log_2(x+3) \] to be defined, \[ x+3\gt 0 \] Thus, \[ x\gt -3 \]
Step 2: Condition for square root in denominator.
Since the square root is in the denominator, \[ x^2+3x+2\gt 0 \] Factorizing: \[ x^2+3x+2=(x+1)(x+2) \] Therefore, \[ (x+1)(x+2)\gt 0 \] This inequality is satisfied when: \[ x\lt -2 \] or \[ x\gt -1 \] Thus, \[ x\in(-\infty,-2)\cup(-1,\infty) \]
Step 3: Find the common interval.
From logarithm condition: \[ x\gt -3 \] From denominator condition: \[ x\in(-\infty,-2)\cup(-1,\infty) \] Taking intersection: \[ (-3,\infty)\cap\left[(-\infty,-2)\cup(-1,\infty)\right] \] we get: \[ (-3,-2)\cup(-1,\infty) \]
Step 4: Final conclusion.
Hence, the domain of the function is \[ \boxed{(-3,-2)\cup(-1,\infty)} \]
For \[ \log_2(x+3) \] to be defined, \[ x+3\gt 0 \] Thus, \[ x\gt -3 \]
Step 2: Condition for square root in denominator.
Since the square root is in the denominator, \[ x^2+3x+2\gt 0 \] Factorizing: \[ x^2+3x+2=(x+1)(x+2) \] Therefore, \[ (x+1)(x+2)\gt 0 \] This inequality is satisfied when: \[ x\lt -2 \] or \[ x\gt -1 \] Thus, \[ x\in(-\infty,-2)\cup(-1,\infty) \]
Step 3: Find the common interval.
From logarithm condition: \[ x\gt -3 \] From denominator condition: \[ x\in(-\infty,-2)\cup(-1,\infty) \] Taking intersection: \[ (-3,\infty)\cap\left[(-\infty,-2)\cup(-1,\infty)\right] \] we get: \[ (-3,-2)\cup(-1,\infty) \]
Step 4: Final conclusion.
Hence, the domain of the function is \[ \boxed{(-3,-2)\cup(-1,\infty)} \]
Was this answer helpful?
0
0
Top AP EAPCET Mathematics Questions
- $D = \left\{x\in\mathbb{R}: f\left(x\right) =\sqrt{\frac{x - \left|x\right|}{x - \left[x\right]}} \text{is defined} \right\}$ and $C$ be the range of the real function $g(x) = \frac{2x}{4 + x^{2}}$. Then $D \cap C$
- $f\left(x\right) = \frac{x}{e^{x}-1} + \frac{x}{2} + 2 \cos^{3} \frac{x}{2} $ on $R-\left\{0\right\} $ is
- Consider the following lists.
List-I List-II (A) $f(x) = \frac{|x+2|}{x+2} , x \ne -2 $ (I) $[\frac{1}{3} , 1 ]$ (B) $(x)=|[x]|,x \in [R$ (II) Z (C) $h(x) = |x - [x]| , x \in [R$ (III) W (D) $f(x) = \frac{1}{2 - \sin 3x} , x \in [R$ (IV) [0, 1) (V) { -1, 1} - If $[x]$ is the greatest integer less than or equal to $x$ and $|x|$ is the modulus of $x$. then the system of three equations $2x + 3 | y | + 5[z] = 0, x + |y| - 2[z] = 4, x + |y| + |z| = 1$ has
- Investigate the values of $\lambda$ and $\mu$ for the system $x+2 y+3 z=6, x+3 y+5 z=9$, $2 x+5 y+\lambda z=\mu$ and match the values in List - I with the items in List - II.
List I List II (A) $\lambda=8, \mu \neq 15$ 1. Infinitely many solutions (B) $\lambda \neq 8, \mu \in R$ 2. No solution (C) $\lambda=8, \mu=15$ 3. Unique solution
View More Questions
Top AP EAPCET Functions Questions
- Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a function defined by \( f(x) = \frac{2x+1}{3} \). If \( \alpha \) is an element in the domain of \( f \) whose image is \( \frac{1}{\alpha} \), then the sum of all possible values of such \( \alpha \) is
- Let \( f(x) = |x| \) and \( g(x) = |x| + a \), \( (a>0) \). For \( 0 \le x \le b \), \( \{(x, y) / g(x) \le y \le f(x)\} \) represents all the points in the interior of
- The range of the function $ f(x) = \begin{cases} 4x - 1, & x>3 \\ x^2 - 2, & -2 \leq x \leq 3 \\ 3x + 4, & x<-2 \end{cases} $ is:
- The domain of the function $ y = f(x) $, where $ x $ and $ y $ are related by $ 2^x + 2^y = 2 $, is:
- If $ \sin x \cosh y = \cos \theta $ and $ \cos x \sinh y = \sin \theta $, then find $ \sin^2 x + \cosh^2 y $.
View More Questions
Top AP EAPCET Questions
- A body is projected from the ground at an angle of $\tan^{-1} \left( \frac{8}{7}\right)$ with the horizontal. The ratio of the maximum height attained by it to its range is
- The four quantum numbers of the valence electron of potassium are :
- A lens forms real and virtual images of an object, when the object is at $u_{1}$ and $u_{2}$ distances respectively. If the size of the virtual image is double that of the real image, then the focal length of the lens is (take, the magnification of the real image as $m$ )
- Two spheres $P$ and $Q$, each of mass $200\, g$ are attached to a string of length one metre as shown in the figure. The string and the spheres are then whirled in a horizontal circle about $O$ at a constant angular speed. The ratio of the tension in the string between $P$ and $Q$ to that of between $P$ and $O$ is $(P$ is at mid-point of the line joining $O$ and $Q$ )
- The volume of $0.02\, M$ acidified permanganate solution required for complete reaction of $60\, mL$ of $0.01\, MI^{-}$ ion solution to form $I_2$ in $mL$ is
View More Questions