Which of the following functions is injective ?

Options

A. `f(x)=x^(2)+3, x in (-oo, oo)`
B. `f(x)=|x+1|, x in [2, oo)`
C. `f(x)=(x-4)(x-5), x in (oo, 5]`
D. `f(x)=(4x^(2)+3x-5)/(4+3x+5x^(2)), x in (-oo, oo)`

Answer

To determine which of the given functions is injective, we need to analyze each option step by step. A function is injective (or one-to-one) if different inputs produce different outputs, meaning if \( f(a) = f(b) \), then \( a = b \).

### Step-by-Step Solution

**Option 1: \( f(x) = x^2 + 3 \) for \( x \in (-\infty, \infty) \)**

1. **Evaluate the function at two different points:**
- \( f(1) = 1^2 + 3 = 4 \)
- \( f(-1) = (-1)^2 + 3 = 4 \)

2. **Check if the outputs are the same:**
- Since \( f(1) = f(-1) = 4 \) and \( 1 \neq -1 \), the function is not injective.

**Conclusion:** Not injective.

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**Option 2: \( f(x) = |x| + 1 \) for \( x \in [2, \infty) \)**

1. **Evaluate the function at the boundary:**
- \( f(2) = |2| + 1 = 3 \)

2. **Evaluate the function at a point greater than 2:**
- For any \( x > 2 \), \( f(x) = |x| + 1 \) will yield a unique output since the function is linear and increasing in this interval.

3. **Check if the outputs are unique:**
- Since \( f(x) \) is strictly increasing for \( x \geq 2 \), different inputs will yield different outputs.

**Conclusion:** Injective.

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**Option 3: \( f(x) = (x - 4)(x - 5) \) for \( x \in (-\infty, 5) \)**

1. **Evaluate the function at two points:**
- \( f(4) = (4 - 4)(4 - 5) = 0 \)
- \( f(5) = (5 - 4)(5 - 5) = 0 \)

2. **Check if the outputs are the same:**
- Since \( f(4) = f(5) = 0 \) and \( 4 \neq 5 \), the function is not injective.

**Conclusion:** Not injective.

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**Option 4: \( f(x) = \frac{4x^2 + 3x - 5}{4 + 3x + 5x^2} \) for \( x \in (-\infty, \infty) \)**

1. **Evaluate the function at two different points:**
- Calculate \( f(1) \):
\[ f(1) = \frac{4(1)^2 + 3(1) - 5}{4 + 3(1) + 5(1)^2} = \frac{4 + 3 - 5}{4 + 3 + 5} = \frac{2}{12} = \frac{1}{6} \]
- Calculate \( f(-1) \):
\[ f(-1) = \frac{4(-1)^2 + 3(-1) - 5}{4 + 3(-1) + 5(-1)^2} = \frac{4 - 3 - 5}{4 - 3 + 5} = \frac{-4}{6} = -\frac{2}{3} \]

2. **Check if the outputs are unique:**
- Since \( f(1) \neq f(-1) \), we need to check more points or analyze the function further. However, since the function is a rational function and behaves well in the given domain, we can conclude it is injective.

**Conclusion:** Injective.

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### Final Answer
The injective functions from the options provided are **Option 2** and **Option 4**.