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

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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**
The function $$ f(x) = 4x + 3 $$ is **one-one** but **not onto**.

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