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Q3:
Find the domain of the function $f(x) = \frac{x^2 + 2x + 1}{x^2 - 8x + 12}$.
Solution :
Step 1: Theoretical Foundation of Domain
The domain of a rational function $f(x) = \frac{P(x)}{Q(x)}$ consists of all real numbers $x \in \mathbb{R}$ such that the denominator $Q(x) \neq 0$. Any value of $x$ that results in $Q(x) = 0$ is excluded from the domain to avoid division by zero, which is undefined in the set of real numbers [Per the definition of functions in $\mathbb{R}$].
Step 2: Identifying the Denominator
Given $f(x) = \frac{x^2 + 2x + 1}{x^2 - 8x + 12}$, the denominator is defined as the quadratic polynomial:
$Q(x) = x^2 - 8x + 12$
Step 3: Solving for Roots of the Denominator
To find the excluded values, we set $Q(x) = 0$ and solve for $x$ using the factorization method [By the Zero Product Property]:
$x^2 - 8x + 12 = 0$
We seek two numbers that multiply to $12$ and add to $-8$. These numbers are $-6$ and $-2$.
$x^2 - 6x - 2x + 12 = 0$
$x(x - 6) - 2(x - 6) = 0$
$(x - 2)(x - 6) = 0$
Therefore, $x - 2 = 0 \implies x = 2$ or $x - 6 = 0 \implies x = 6$.
Step 4: Establishing the Domain Interval
The function $f(x)$ is defined for all real numbers except for the set of points where the denominator vanishes. Thus, the domain is the set of all real numbers excluding $\{2, 6\}$.
Step 5: Final Set-Builder Notation
Expressing the result in standard set notation:
$Domain(f) = \mathbb{R} \setminus \{2, 6\}$
Alternatively, in interval notation:
$Domain(f) = (-\infty, 2) \cup (2, 6) \cup (6, \infty)$
Final Solution: The domain of the function is the set of all real numbers except $2$ and $6$, represented as $\mathbb{R} - \{2, 6\}$.
More Questions from Class 11 Mathematics Relations and Functions Miscellaneous Exercise on Chapter 2
- Q1: The relation $f$ is defined by $f(x) = \begin{cases} x^2, & 0 \le x \le 3 \\ 3x, & 3 \le x \le 10 \end{cases}$. The relation $g$ is defined by $g(x) = \begin{cases} x^2, & 0 \le x \le 2 \\ 3x, & 2 \le x \le 10 \end{cases}$. Show that $f$ is a function and $g$ is not a function.
- Q2: If $f(x) = x^2$, find $\frac{f(1.1) - f(1)}{1.1 - 1}$.
- Q4: Find the domain and the range of the real function $f$ defined by $f(x) = \sqrt{x - 1}$.
- Q5: Find the domain and the range of the real function $f$ defined by $f(x) = |x – 1|$.
- Q6: Let $f = \{\left(x, \frac{x^2}{1+x^2}\right) : x \in \mathbb{R}\}$ be a function from $\mathbb{R}$ into $\mathbb{R}$. Determine the range of $f$.
- Q7: Let $f, g : \mathbb{R} \to \mathbb{R}$ be defined, respectively by $f(x) = x + 1$, $g(x) = 2x – 3$. Find $f + g$, $f – g$ and $\frac{f}{g}$.
- Q8: Let $f = \{(1,1), (2,3), (0,–1), (–1, –3)\}$ be a function from $\mathbb{Z}$ to $\mathbb{Z}$ defined by $f(x) = ax + b$, for some integers $a, b$. Determine $a, b$.
- Q9(i): Let $R$ be a relation from $\mathbb{N}$ to $\mathbb{N}$ defined by $R = \{(a, b) : a, b \in \mathbb{N} \text{ and } a = b^2\}$. Are the following true? (i) $(a,a) \in R$, for all $a \in \mathbb{N}$. Justify your answer in each case.
- Q9(ii): Let $R$ be a relation from $\mathbb{N}$ to $\mathbb{N}$ defined by $R = \{(a, b) : a, b \in \mathbb{N} \text{ and } a = b^2\}$. Are the following true? (ii) $(a,b) \in R$, implies $(b,a) \in R$. Justify your answer in each case.
- Q9(iii): Let $R$ be a relation from $\mathbb{N}$ to $\mathbb{N}$ defined by $R = \{(a, b) : a, b \in \mathbb{N} \text{ and } a = b^2\}$. Are the following true? (iii) $(a,b) \in R, (b,c) \in R$ implies $(a,c) \in R$. Justify your answer in each case.
- Q10(i): Let $A =\{1,2,3,4\}$, $B = \{1,5,9,11,15,16\}$ and $f = \{(1,5), (2,9), (3,1), (4,5), (2,11)\}$. Are the following true? (i) $f$ is a relation from $A$ to $B$. Justify your answer in each case.
- Q10(ii): Let $A =\{1,2,3,4\}$, $B = \{1,5,9,11,15,16\}$ and $f = \{(1,5), (2,9), (3,1), (4,5), (2,11)\}$. Are the following true? (ii) $f$ is a function from $A$ to $B$. Justify your answer in each case.
- Q11: Let $f$ be the subset of $\mathbb{Z} \times \mathbb{Z}$ defined by $f = \{(ab, a + b) : a, b \in \mathbb{Z}\}$. Is $f$ a function from $\mathbb{Z}$ to $\mathbb{Z}$? Justify your answer.
- Q12: Let $A = \{9,10,11,12,13\}$ and let $f : A \to \mathbb{N}$ be defined by $f(n) = \text{the highest prime factor of } n$. Find the range of $f$.
CBSE Solutions for Class 11 Mathematics Relations and Functions
Chapters in CBSE - Class 11 Mathematics
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