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Q4:
Find the domain and the range of the real function $f$ defined by $f(x) = \sqrt{x - 1}$.
Solution :
Step 1: Establishing the Domain Condition
The function is defined as $f(x) = \sqrt{x - 1}$. In the set of real numbers $\mathbb{R}$, the square root operation $\sqrt{u}$ is defined if and only if the radicand $u \geq 0$.
Setting the condition for the function $f(x)$:
$x - 1 \geq 0$ [By the definition of real-valued functions requiring non-negative radicands for square roots]
$x \geq 1$ [By the Addition Property of Inequality]
Therefore, the domain of $f$ is the interval $[1, \infty)$.
Step 2: Determining the Range
Let $y = f(x) = \sqrt{x - 1}$. By the definition of the principal square root function, the output $y$ must always be non-negative, regardless of the value of $x$ within the domain.
- At the boundary of the domain, $x = 1$, $f(1) = \sqrt{1 - 1} = 0$.
- As $x \to \infty$, $f(x) \to \infty$.
Since $x - 1 \geq 0$, it follows that $\sqrt{x - 1} \geq 0$. Consequently, $y \geq 0$ [By the definition of the range of the principal square root function].
The range of $f$ is the interval $[0, \infty)$.
Step 3: Visual Representation
Step 4: Summary of Findings
| Feature | Interval Notation |
|---|---|
| Domain | $[1, \infty)$ |
| Range | $[0, \infty)$ |
Final Solution: The domain of the function $f(x) = \sqrt{x - 1}$ is $[1, \infty)$ and the range is $[0, \infty)$.
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 $f(x) = \frac{x^2 + 2x + 1}{x^2 - 8x + 12}$.
- 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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