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Q2(v):
Check the injectivity and surjectivity of the following functions: (v) $f : Z \to Z$ given by $f(x) = x^3$
Solution :
Step 1: Formal Definition of the Function
The function is defined as $f: \mathbb{Z} \to \mathbb{Z}$, where $\mathbb{Z}$ represents the set of all integers ($\dots, -2, -1, 0, 1, 2, \dots$). The rule of correspondence is $f(x) = x^3$.
Step 2: Checking for Injectivity (One-to-One)
A function $f: A \to B$ is injective if $f(x_1) = f(x_2) \implies x_1 = x_2$ for all $x_1, x_2 \in A$. [Definition of Injective Mapping]
Let $x_1, x_2 \in \mathbb{Z}$ such that $f(x_1) = f(x_2)$.
Substituting the function rule: $x_1^3 = x_2^3$.
Taking the cube root on both sides: $x_1 = x_2$.
Since the cube root of an integer is unique in the domain of real numbers and preserves the integer property, $x_1 = x_2$ is the only possible solution. Therefore, $f$ is injective.
Step 3: Checking for Surjectivity (Onto)
A function $f: A \to B$ is surjective if for every $y \in B$ (codomain), there exists at least one $x \in A$ (domain) such that $f(x) = y$. [Definition of Surjective Mapping]
Let $y \in \mathbb{Z}$ be an arbitrary element in the codomain. For the function to be surjective, $x^3 = y$ must have an integer solution $x \in \mathbb{Z}$ for every $y \in \mathbb{Z}$.
Consider $y = 2$. Since $y \in \mathbb{Z}$ (the codomain), we check if there exists an $x \in \mathbb{Z}$ such that $x^3 = 2$.
$x = \sqrt[3]{2}$.
However, $\sqrt[3]{2} \approx 1.2599$, which is not an integer. Since $\sqrt[3]{2} \notin \mathbb{Z}$, there is no pre-image for the element $2$ in the domain.
Step 4: Conclusion of Mapping Properties
The element $y=2$ (and many others like $y=3, 4, 5, \dots$) has no corresponding $x \in \mathbb{Z}$ such that $f(x)=y$. Thus, the range of $f$ is not equal to the codomain $\mathbb{Z}$.
Final Solution: The function $f(x) = x^3$ is injective but not 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 $f(x) = x^3$
- 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.
- 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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