Question:
The domain of the function $f \left(x\right)=\frac{\log_{2}\left(x+3\right)}{x^{2}+3x+2}$ is
The domain of the function $f \left(x\right)=\frac{\log_{2}\left(x+3\right)}{x^{2}+3x+2}$ is
Updated On: Jun 7, 2024
- $R - \{ -1,-2 \}$
- $R -\{ -1,-2 ,0 \}$
- $(-3 ,-1)$ $\cup$ $(-1, \infty)$
- $(-3, \infty) - \{-1,-2\}$
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The Correct Option is D
Solution and Explanation
Given function is $f(x)=\frac{\log _{2}(x+3)}{x^{2}+3 x+2}$
Here, for existence of log
$x+3>\,0$
$ \Rightarrow x >\, -3$
$\Rightarrow\, x \in(-3, \infty)$
and for existence of $f(x)$,
$ x^{2}+3 x+ 2 \, \neq 0 $
$\Rightarrow \,(x+1)(x+2) \neq 0 $
$\Rightarrow \, x \neq-1,-2$
Hence, required domain of $f(x)$ is
$(-3, \infty)-\{-1,-2\}$
Here, for existence of log
$x+3>\,0$
$ \Rightarrow x >\, -3$
$\Rightarrow\, x \in(-3, \infty)$
and for existence of $f(x)$,
$ x^{2}+3 x+ 2 \, \neq 0 $
$\Rightarrow \,(x+1)(x+2) \neq 0 $
$\Rightarrow \, x \neq-1,-2$
Hence, required domain of $f(x)$ is
$(-3, \infty)-\{-1,-2\}$
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Concepts Used:
Relations and functions
A relation R from a non-empty set B is a subset of the cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.
A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B. In other words, no two distinct elements of B have the same pre-image.
Representation of Relation and Function
Relations and functions can be represented in different forms such as arrow representation, algebraic form, set-builder form, graphically, roster form, and tabular form. Define a function f: A = {1, 2, 3} → B = {1, 4, 9} such that f(1) = 1, f(2) = 4, f(3) = 9. Now, represent this function in different forms.
- Set-builder form - {(x, y): f(x) = y2, x ∈ A, y ∈ B}
- Roster form - {(1, 1), (2, 4), (3, 9)}
- Arrow Representation
