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Q12:
Let $f : R \to R$ be defined as $f(x) = 3x$. Choose the correct answer.
Let $f : R \to R$ be defined as $f(x) = 3x$. Choose the correct answer.
Solution :
Step 1: Analyzing the Function Definition
The function $f: \mathbb{R} \to \mathbb{R}$ is defined by the mapping rule $f(x) = 3x$. The domain and codomain are both the set of real numbers ($\mathbb{R}$).
Step 2: Testing for Injectivity (One-to-One Function)
A function $f$ is injective if $f(x_1) = f(x_2)$ implies $x_1 = x_2$ for all $x_1, x_2$ in the domain [Definition of Injective Mapping].
- Let $x_1, x_2 \in \mathbb{R}$ such that $f(x_1) = f(x_2)$.
- Substituting the given rule: $3x_1 = 3x_2$.
- Dividing both sides by $3$ (a non-zero scalar): $x_1 = x_2$.
Since the condition $f(x_1) = f(x_2) \implies x_1 = x_2$ is satisfied, the function $f$ is one-to-one.
Step 3: Testing for Surjectivity (Onto Function)
A function $f: A \to B$ is surjective if for every element $y \in B$, there exists at least one element $x \in A$ such that $f(x) = y$ [Definition of Surjective Mapping].
- Let $y$ be any arbitrary real number in the codomain $\mathbb{R}$.
- Set $f(x) = y$, which implies $3x = y$.
- Solving for $x$: $x = \frac{y}{3}$.
- Since $y \in \mathbb{R}$, then $\frac{y}{3}$ must also be a real number. Thus, for every $y \in \mathbb{R}$, there exists an $x = \frac{y}{3} \in \mathbb{R}$ such that $f(\frac{y}{3}) = 3(\frac{y}{3}) = y$.
Since the codomain equals the range, the function $f$ is onto.
Step 4: Visualizing the Mapping
Step 5: Synthesizing the Properties
| Property | Status | Logical Deduction |
|---|---|---|
| Injective | Yes | $3x_1 = 3x_2 \implies x_1 = x_2$ |
| Surjective | Yes | $\forall y \in \mathbb{R}, \exists x = y/3 \in \mathbb{R}$ |
Final Solution: The function $f(x) = 3x$ is both one-to-one and onto (bijective).
More Questions from Class 12 Mathematics Relations and Functions EXERCISE 1.2
- Q1: Show that the function $f : R_{*} \to R_{*}$ defined by $f(x) = \frac{1}{x}$ is one-one and onto, where $R_{*}$ is the set of all non-zero real numbers. Is the result true, if the domain $R_{*}$ is replaced by $N$ with co-domain being same as $R_{*}$?
- Q2(i): Check the injectivity and surjectivity of the following functions: (i) $f : N \to N$ given by $f(x) = x^2$
- Q2(ii): Check the injectivity and surjectivity of the following functions: (ii) $f : Z \to Z$ given by $f(x) = x^2$
- Q2(iii): Check the injectivity and surjectivity of the following functions: (iii) $f : R \to R$ given by $f(x) = x^2$
- Q2(iv): Check the injectivity and surjectivity of the following functions: (iv) $f : N \to N$ given by $f(x) = x^3$
- Q2(v): Check the injectivity and surjectivity of the following functions: (v) $f : Z \to Z$ given by $f(x) = x^3$
- Q3: Prove that the Greatest Integer Function $f : R \to R$, given by $f(x) = [x]$, is neither one-one nor onto, where $[x]$ denotes the greatest integer less than or equal to $x$.
- Q4: Show that the Modulus Function $f : R \to R$, given by $f(x) = |x|$, is neither one-one nor onto, where $|x|$ is $x$, if $x$ is positive or 0 and $|x|$ is $– x$, if $x$ is negative.
- Q5: Show that the Signum Function $f : R \to R$, given by $f(x) = \begin{cases} 1 & \text{if } x > 0 \\ 0 & \text{if } x = 0 \\ -1 & \text{if } x < 0 \end{cases}$ is neither one-one nor onto.
- Q6: Let $A = \{1, 2, 3\}$, $B = \{4, 5, 6, 7\}$ and let $f = \{(1, 4), (2, 5), (3, 6)\}$ be a function from $A$ to $B$. Show that $f$ is one-one.
- Q7(i): In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer. (i) $f : R \to R$ defined by $f(x) = 3 – 4x$
- Q7(ii): In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer. (ii) $f : R \to R$ defined by $f(x) = 1 + x^2$
- Q8: Let $A$ and $B$ be sets. Show that $f : A \times B \to B \times A$ such that $f(a, b) = (b, a)$ is bijective function.
- Q9: Let $f : N \to N$ be defined by $f(n) = \begin{cases} \frac{n+1}{2} & \text{if } n \text{ is odd} \\ \frac{n}{2} & \text{if } n \text{ is even} \end{cases}$ for all $n \in N$. State whether the function $f$ is bijective. Justify your answer.
- Q10: Let $A = R – \{3\}$ and $B = R – \{1\}$. Consider the function $f : A \to B$ defined by $f(x) = \frac{x-2}{x-3}$. Is $f$ one-one and onto? Justify your answer.
- Q11: Let $f : R \to R$ be defined as $f(x) = x^4$. Choose the correct answer.
CBSE Solutions for Class 12 Mathematics Relations and Functions
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