Which option (s) is/are ture ?

Options

A. `f :R to R, f (x) =e ^(|x|)-e ^(-x)` is many-one into function
B. `f : R to R, f (x)=2x+|sin x|` is one-one onto
C. `f:R to R, f (x)=(x^(2) +4x+30)/( x ^(2) -8x+18)` is many-one onto
D. `f :R to R, f (x) =(2x ^(2) -x+5)/( 7x^(2) +2x+10)` is many-one into

Answer

To solve the problem, we need to analyze the given function options one by one to determine which ones are true based on their properties (one-one, onto, etc.).

### Step-by-Step Solution:

**Step 1: Analyze Option A**
- Function: \( f(x) = e^{|x|} - e^{-x} \)
- For \( x < 0 \):
- \( |x| = -x \)
- Therefore, \( f(x) = e^{-x} - e^{x} \)
- This function is decreasing and takes the value 0 when \( x = -1 \) and \( x = -2 \).
- For \( x \geq 0 \):
- \( f(x) = 2 \sinh(x) \) which is positive and increasing.
- **Conclusion**: The function is not onto since its range does not cover all of \( \mathbb{R} \) (it only covers \( [0, \infty) \)). Thus, Option A is **true**.

**Step 2: Analyze Option B**
- Function: \( f(x) = 2x + |\sin x| \)
- The derivative \( f'(x) = 2 + \frac{\sin x}{|\sin x|} \cos x \) is always positive.
- This indicates that \( f(x) \) is strictly increasing.
- The range of this function is \( (-\infty, \infty) \) and matches the codomain.
- **Conclusion**: The function is one-one and onto. Thus, Option B is **true**.

**Step 3: Analyze Option C**
- Function: \( f(x) = \frac{x^2 + 4x + 30}{x^2 - 8x + 18} \)
- To check if it is one-one, we find the derivative using the quotient rule:
- \( f'(x) = \frac{(2x + 4)(x^2 - 8x + 18) - (x^2 + 4x + 30)(2x - 8)}{(x^2 - 8x + 18)^2} \)
- The numerator can be positive or negative, indicating that the function is not necessarily one-one.
- Additionally, for \( y = 0 \), there is no \( x \) that satisfies this, indicating it is not onto.
- **Conclusion**: Option C is **false**.

**Step 4: Analyze Option D**
- Function: \( f(x) = \frac{2x^2 - x + 5}{7x^2 + 2x + 10} \)
- Again, we find the derivative using the quotient rule:
- \( f'(x) = \frac{(4x - 1)(7x^2 + 2x + 10) - (2x^2 - x + 5)(14x + 2)}{(7x^2 + 2x + 10)^2} \)
- The sign of the derivative can vary, indicating that it is not one-one.
- Also, for \( y = 0 \), there is no \( x \) that satisfies this, indicating it is not onto.
- **Conclusion**: Option D is **false**.

### Final Conclusion:
- The true options are **A and B**.