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 functions to determine their nature (one-to-one, many-to-one, onto, or into). Let's go through each function step by step.

### Step 1: Analyze the first function
**Function:** \( f(x) = e^{|x|} - e^{-x} \)

1. **For \( x < 0 \):**
\[ f(x) = e^{-x} - e^{x} = -2e^{-x} \]
This is a decreasing function since \( e^{-x} \) is always positive and decreases as \( x \) decreases.

2. **For \( x > 0 \):**
\[ f(x) = e^{x} - e^{-x} \]
This is an increasing function since both \( e^{x} \) and \( e^{-x} \) are positive and \( e^{x} \) increases faster than \( e^{-x} \).

3. **Conclusion:**
- The function is many-to-one for \( x < 0 \) and one-to-one for \( x > 0 \).
- The range is \( [0, \infty) \), which means it is into (not onto).