If $f(x)=log_{e}(x+\sqrt{1+x^{2}})$, then $f^{-1}(x)$ is equal to:

Question

Accepted Answer

To determine the inverse function $ f^{-1}(x) $ for the given function $ f(x) = \log_e(x + \sqrt{1 + x^2}) $, we need to find a function $ y $ such that $ f(y) = x $.

### Step-by-step Explanation:

1. **Start with the given function:**
   $$
   f(x) = \log_e(x + \sqrt{1 + x^2})
   $$

2. **Set $ f(y) = x $:**
   $$
   \log_e(y + \sqrt{1 + y^2}) = x
   $$

3. **Exponentiate both sides to eliminate the logarithm:**
   $$
   y + \sqrt{1 + y^2} = e^x
   $$

4. **Isolate $ y $:**
   - Square both sides to remove the square root:
     $$
     (y + \sqrt{1 + y^2})^2 = e^{2x}
     $$
   - Expand the left side:
     $$
     y^2 + 2y\sqrt{1 + y^2} + 1 + y^2 = e^{2x}
     $$
   - Simplify:
     $$
     2y^2 + 2y\sqrt{1 + y^2} + 1 = e^{2x}
     $$
   - Rearrange to isolate the square root:
     $$
     2y\sqrt{1 + y^2} = e^{2x} - 2y^2 - 1
     $$

5. **Solve for $ y $:**
   - Recognize that $ y = \sinh(x) $ satisfies the equation $ y + \sqrt{1 + y^2} = e^x $ because:
     $$
     \sinh(x) = \frac{e^x - e^{-x}}{2}
     $$
   - Therefore, $ y = \frac{e^x - e^{-x}}{2} $.

Thus, the inverse function $ f^{-1}(x) $ is:
$$
f^{-1}(x) = \frac{e^x - e^{-x}}{2}
$$

### Explanation of Incorrect Options:

- **Option 2: $\frac{e^x + e^{-x}}{2}$**
  - This represents the hyperbolic cosine function, $\cosh(x)$, which does not satisfy the equation derived from the inverse process. The function $\cosh(x)$ does not match the requirement of the inverse function for $ f(x) $.

- **Option 3: $\frac{e^x - e^{-x}}{e^x + e^{-x}}$**
  - This expression simplifies to the hyperbolic tangent function, $\tanh(x)$. The function $\tanh(x)$ does not satisfy the equation derived for the inverse function.

- **Option 4: $\frac{e^x + e^{-x}}{e^x - e^{-x}}$**
  - This expression does not correspond to any standard hyperbolic function that would satisfy the inverse relationship for the given $ f(x) $.

In conclusion, Option 1 is the correct answer because it correctly represents the inverse function $ f^{-1}(x) $ as derived from the original function $ f(x) = \log_e(x + \sqrt{1 + x^2}) $.