Relations \& Functions 1. A function \( f: R_{+} \rightarrow R \) (where \( R_{+} \)is the set of all non-negative real numbers) defined by \( f(x)=4 x+3 \) is : (2024) (A) one-one but not onto (B) onto but not one-one (C) both one-one and onto (D) neither one-one nor onto Ans. (A) one-one but not onto

Let's analyze the function \( f: \mathbb{R}_{+} \rightarrow \mathbb{R} \) defined by \( f(x) = 4x + 3 \), where \( \mathbb{R}_{+} \) represents the set of all non-negative real numbers (\( x \geq 0 \)). ### 1. **One-One (Injective)** A function is **one-one** or **injective** if distinct inputs produce distinct outputs. - **For \( f(x) = 4x + 3 \):** - Suppose \( f(a) = f(b) \). - Then, \( 4a + 3 = 4b + 3 \). - Subtracting 3 from both sides: \( 4a = 4b \). - Dividing both sides by 4: \( a = b \). Since \( a = b \), the function \( f \) is **injective**. ### 2. **Onto (Surjective)** A function is **onto** or **surjective** if every element in the codomain has a pre-image in the domain. - **Codomain:** \( \mathbb{R} \) (all real numbers). - **Domain:** \( \mathbb{R}_{+} \) (all non-negative real numbers). - **For \( f(x) = 4x + 3 \):** - The smallest value \( f(x) \) can take is when \( x = 0 \): \( f(0) = 3 \). - As \( x \) increases, \( f(x) \) increases without bound. - Therefore, \( f(\mathbb{R}_{+}) = [3, \infty) \). Since the codomain \( \mathbb{R} \) includes all real numbers below 3 as well, there are real numbers (specifically, all \( y < 3 \)) that **do not** have a corresponding \( x \) in \( \mathbb{R}_{+} \) such that \( f(x) = y \). Thus, the function \( f \) is **not surjective**. ### **Conclusion** - **Injective:** Yes - **Surjective:** No Therefore, the correct answer is: **(A) one-one but not onto**