Question:
Let $f: R \rightarrow R$ be defined by $f(x)=\frac{x^{2}-3 x-6}{x^{2}+2 x+4}$. Then which of the following statements is(are) TRUE?
Let $f: R \rightarrow R$ be defined by $f(x)=\frac{x^{2}-3 x-6}{x^{2}+2 x+4}$. Then which of the following statements is(are) TRUE?
Updated On: Oct 6, 2024
- $f$ is decreasing in the interval $(-2,-1)$
- $f$ is increasing in the interval $(1,2)$
- $f$ is onto
- Range of $f$ is $\left[-\frac{3}{2}, 2\right]$
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The Correct Option is A, B
Solution and Explanation
\(f(x) = \frac{{x^2 - 3x - 6}}{{x^2 + 2x + 4}}\)
\(⇒\) \(f’(x) =\) \(\frac{{5\times(x+4)}}{{(x^2 + 2x + 4)^2}}\)
\(⇒ f(x)\) has local maxima at x = -4 and minima at x = 0
Range of \(f(x)\) is \(\left[-\frac{3}{2}, \frac{11}{6}\right]\)
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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.
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- Arrow Representation
