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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.
Solution :
Step 1: Formal Definition of One-One (Injective) Mapping
A function $f: X \to Y$ is defined as one-one if and only if for all $x_1, x_2 \in X$, $f(x_1) = f(x_2)$ implies $x_1 = x_2$ [Definition of Injective Mapping].
Step 2: Testing for One-One Property
Let us select two distinct values from the domain $R$, such as $x_1 = 2$ and $x_2 = 3$.
- According to the definition of the Signum Function, since $2 > 0$, $f(2) = 1$.
- Since $3 > 0$, $f(3) = 1$.
Here, $f(2) = f(3)$ but $2 \neq 3$. [By contradiction of the injective criterion].
Since multiple distinct elements in the domain map to the same image (1), the function is not one-one.
Step 3: Formal Definition of Onto (Surjective) Mapping
A function $f: X \to Y$ is onto if for every $y \in Y$, there exists at least one $x \in X$ such that $f(x) = y$. In this case, the codomain is the set of all real numbers, $R$ [Definition of Surjective Mapping].
Step 4: Testing for Onto Property
The range of the signum function is restricted to the set $\{-1, 0, 1\}$.
Consider any real number $y \in R$ that is not in the set $\{-1, 0, 1\}$, for example, $y = 2$.
There exists no $x \in R$ such that $f(x) = 2$, because the output of $f(x)$ is strictly limited to $\{1, 0, -1\}$ by definition. Therefore, the codomain $R$ is not equal to the range $\{-1, 0, 1\}$.
Since there are elements in the codomain with no pre-image in the domain, the function is not onto.
Visual Representation of the Signum Function
Final Solution: The Signum function $f(x)$ is neither one-one because $f(2) = f(3)$, nor onto because the range $\{-1, 0, 1\} \subset R$ does not cover the codomain $R$.
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.
- 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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