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Q5:
Find the domain and the range of the real function $f$ defined by $f(x) = |x – 1|$.
Solution :
Step 1: Theoretical Definition of the Function
The function $f(x) = |x - 1|$ is a modulus (absolute value) function. By the fundamental definition of the absolute value operator, the function maps any input $x$ to its non-negative magnitude, defined piecewise as:
$f(x) = \begin{cases} -(x - 1), & \text{if } x - 1 < 0 \\ (x - 1), & \text{if } x - 1 \ge 0 \end{cases}$This simplifies to:
$f(x) = \begin{cases} 1 - x, & \text{if } x < 1 \\ x - 1, & \text{if } x \ge 1 \end{cases}$Step 2: Determination of the Domain
The domain of a real-valued function is the set of all possible real values of $x$ for which $f(x)$ is defined. Since the expression $(x - 1)$ is a polynomial, it is defined for all real numbers $x \in \mathbb{R}$. There are no values of $x$ that lead to division by zero, square roots of negative numbers, or undefined logarithmic arguments. [By the property of polynomial functions in $\mathbb{R}$]
Domain: $\text{Dom}(f) = (-\infty, \infty)$ or $\mathbb{R}$.
Step 3: Determination of the Range
The range is the set of all resulting values $f(x)$ for all $x \in \text{Domain}$. By the definition of the modulus function, $|x - 1| \ge 0$ for all $x \in \mathbb{R}$. [By the non-negativity axiom of the absolute value].
As $x$ varies from $-\infty$ to $\infty$, $x-1$ also varies from $-\infty$ to $\infty$. The absolute value of any real number in $(-\infty, \infty)$ yields the set of all non-negative real numbers $[0, \infty)$.
Range: $\text{Ran}(f) = [0, \infty)$.
Step 4: Visual Representation
Summary Table of Functional Properties
| Property | Result |
|---|---|
| Domain | $(-\infty, \infty)$ |
| Range | $[0, \infty)$ |
| Vertex Point | $(1, 0)$ |
Final Solution: The domain of the function $f(x) = |x - 1|$ is $\mathbb{R}$ (all real numbers), and the range is $[0, \infty)$ (all non-negative real numbers).
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 $\frac{f(1.1) - f(1)}{1.1 - 1}$.
- Q3: Find the domain of the function $f(x) = \frac{x^2 + 2x + 1}{x^2 - 8x + 12}$.
- Q4: Find the domain and the range of the real function $f$ defined by $f(x) = \sqrt{x - 1}$.
- 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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Other Subjects in CBSE - Class 11
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