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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.
Solution :
Step 1: Analyzing Injectivity (One-to-One Property)
A function $f: A \to B$ is injective if $f(n_1) = f(n_2) \implies n_1 = n_2$. We evaluate this by testing values for $n \in \mathbb{N}$.
- Consider $n_1 = 1$ (odd): $f(1) = \frac{1+1}{2} = 1$.
- Consider $n_2 = 2$ (even): $f(2) = \frac{2}{2} = 1$.
[By Definition of Injectivity]: Since $f(1) = f(2) = 1$ but $1 \neq 2$, the function $f$ fails the condition for injectivity. Therefore, $f$ is not one-to-one.
Step 2: Analyzing Surjectivity (Onto Property)
A function $f: A \to B$ is surjective if for every $y \in \text{Codomain}$, there exists an $x \in \text{Domain}$ such that $f(x) = y$. Here, the codomain is $\mathbb{N}$.
Case 1: If $y$ is odd, let $n = 2y - 1$. Since $y \geq 1$, $n$ is an odd natural number.
$f(2y-1) = \frac{(2y-1)+1}{2} = \frac{2y}{2} = y$.
Case 2: If $y$ is even, let $n = 2y$. Since $y \geq 1$, $n$ is an even natural number.
$f(2y) = \frac{2y}{2} = y$.
[By Definition of Surjectivity]: Since every element in the codomain $\mathbb{N}$ has a pre-image in the domain $\mathbb{N}$, the function $f$ is onto.
Visual Representation of Mapping
Step 3: Conclusion on Bijectivity
A function is defined as bijective if and only if it is both injective and surjective.
[Per the Definition of Bijective Functions]: Since $f$ is surjective but not injective, it cannot be bijective.
Final Solution: The function $f$ is not bijective because it is not injective (many-to-one), even though it is surjective.
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 $β x$, 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 $f(x) = 3 β 4x$
- 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.
- Q10: Let $A = R β \{3\}$ and $B = R β \{1\}$. 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.
- Q12: Let $f : R \to R$ be defined as $f(x) = 3x$. Choose the correct answer.
CBSE Solutions for Class 12 Mathematics Relations and Functions
Chapters in CBSE - Class 12 Mathematics
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