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Calculate inverse, domain inverse and range inverse. 

To find the inverse of the given function \( f(x) = \sqrt{\frac{e^x - 7}{4 - e^x}} \) and determine its domain and range, follow these steps:

### 1. Finding the Inverse Function

Let's denote the function by \( y = f(x) \). This means:
\[ y = \sqrt{\frac{e^x - 7}{4 - e^x}} \]

To find the inverse, solve for \( x \) in terms of \( y \):

1. Square both sides to eliminate the square root:
\[ y^2 = \frac{e^x - 7}{4 - e^x} \]

2. Cross-multiply to simplify:
\[ y^2 (4 - e^x) = e^x - 7 \]

3. Distribute \( y^2 \):
\[ 4y^2 - y^2 e^x = e^x - 7 \]

4. Collect terms involving \( e^x \):
\[ 4y^2 + 7 = e^x + y^2 e^x \]

5. Factor out \( e^x \) on the right-hand side:
\[ 4y^2 + 7 = e^x (1 + y^2) \]

6. Solve for \( e^x \):
\[ e^x = \frac{4y^2 + 7}{1 + y^2} \]

7. Take the natural logarithm of both sides to solve for \( x \):
\[ x = \ln\left(\frac{4y^2 + 7}{1 + y^2}\right) \]

So, the inverse function \( f^{-1}(x) \) is:
\[ f^{-1}(x) = \ln\left(\frac{4x^2 + 7}{1 + x^2}\right) \]

### 2. Domain and Range of the Inverse Function

#### Domain of the Inverse Function \( f^{-1}(x) \)

The domain of \( f^{-1}(x) \) is the range of the original function \( f(x) \).

For \( f(x) \):
\[ y = \sqrt{\frac{e^x - 7}{4 - e^x}} \]

- The expression under the square root must be non-negative:
  \[ \frac{e^x - 7}{4 - e^x} \geq 0 \]
  - For \( \frac{e^x - 7}{4 - e^x} \geq 0 \):
    - \( e^x - 7 \geq 0 \) and \( 4 - e^x \leq 0 \)
    - \( e^x \geq 7 \) and \( e^x \leq 4 \), which is not possible simultaneously.

  - We need to check for valid intervals:
    - \( e^x - 7 \leq 0 \) and \( 4 - e^x \geq 0 \)
    - \( e^x \leq 7 \) and \( e^x \leq 4 \), so \( 7 \geq e^x \geq 4 \)
    - \( x \leq \ln(7) \) and \( x \leq \ln(4) \)

So, \( x \in (\ln(7), \ln(4)) \).

#### Range of the Inverse Function \( f^{-1}(x) \)

The range of \( f^{-1}(x) \) is the domain of the original function \( f(x) \).

Since the original function \( f(x) = \sqrt{\frac{e^x - 7}{4 - e^x}} \) involves a square root, the output is non-negative. Hence, the range is:
\[ y \geq 0 \]

So, the range of \( f^{-1}(x) \) is:
\[ (-\infty, 0] \]

### Summary

1. **Inverse function:**
   \[
   f^{-1}(x) = \ln\left(\frac{4x^2 + 7}{1 + x^2}\right)
   \]

2. **Domain of the inverse function:**
   \[
   (-\infty, 0]
   \]

3. **Range of the inverse function:**
   \[
   (-\infty, \ln(4))
   \]

Would you like more details or have any questions?

Here are 5 questions you can ask:
1. How do you find the domain and range of a function?
2. Can you explain how to derive the inverse of a function step by step?
3. What are some examples of functions and their inverses?
4. How do you verify if two functions are inverses of each other?
5. What are the properties of exponential and logarithmic functions?

**Tip:** Always check if a function is one-to-one before finding its inverse. A function must be one-to-one to have an inverse.

Learn how to calculate the inverse, domain inverse, and range inverse of a function step-by-step. This problem involves determining the inverse function of f(x), finding its domain, and identifying its range. Explore exponential and logarithmic functions in detail. Suitable for advanced mathematics students.

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