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Q11:
Let $f : R \to R$ be defined as $f(x) = x^4$. Choose the correct answer.
Let $f : R \to R$ be defined as $f(x) = x^4$. Choose the correct answer.
Solution :
Step 1: Analyzing the Injectivity (One-to-One Property)
A function $f: A \to B$ is injective (one-to-one) if and only if $f(x_1) = f(x_2) \implies x_1 = x_2$ for all $x_1, x_2 \in A$. [Per the Definition of Injective Functions].
Given $f(x) = x^4$, let us test two distinct values from the domain $R$, such as $x_1 = 1$ and $x_2 = -1$.
- $f(1) = (1)^4 = 1$
- $f(-1) = (-1)^4 = 1$
Since $f(1) = f(-1)$ but $1 \neq -1$, the function fails the horizontal line test. [By the failure of the Injectivity Criterion].
Conclusion: The function $f(x) = x^4$ is not one-to-one.
Step 2: Analyzing the Surjectivity (Onto Property)
A function $f: R \to R$ is surjective (onto) if and only if for every $y \in R$ (codomain), there exists an $x \in R$ (domain) such that $f(x) = y$. [Per the Definition of Surjective Functions].
For $f(x) = x^4$, since $x$ is raised to an even power, the output $f(x)$ will always be non-negative for any real input $x$. That is, $f(x) \geq 0$ for all $x \in R$.
Consider a value $y = -2$ in the codomain $R$. There is no real number $x$ such that $x^4 = -2$, because the fourth power of any real number cannot be negative. [By the property of Real Numbers and Exponents].
Conclusion: The function $f(x) = x^4$ is not onto.
Step 3: Visual Representation of the Function
Step 4: Comparative Summary
| Property | Status | Reason |
|---|---|---|
| One-to-One | No | $f(1) = f(-1) = 1$ |
| Onto | No | Range $\subset$ Codomain ($[0, \infty) \subset R$) |
Final Solution: The function $f(x) = x^4$ is neither one-to-one nor onto.
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.
- 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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