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The given function is \( f(x) = e^{2x + 1} \).

### (a) Determine \( f^{-1}(x) \), the inverse of the function.

To find the inverse, follow these steps:

1. Start with the function \( y = e^{2x + 1} \).
2. Switch \( x \) and \( y \) to find the inverse: 
   \[
   x = e^{2y + 1}
   \]
3. Take the natural logarithm (ln) of both sides to remove the exponent:
   \[
   \ln(x) = 2y + 1
   \]
4. Solve for \( y \):
   \[
   \ln(x) - 1 = 2y
   \]
   \[
   y = \frac{\ln(x) - 1}{2}
   \]
Thus, the inverse function is:
\[
f^{-1}(x) = \frac{\ln(x) - 1}{2}
\]

### (b) For the function \( f^{-1}(x) \), determine:
#### (i) The domain:
Since the original function \( f(x) = e^{2x+1} \) is an exponential function, its range is \( (0, \infty) \). Thus, for the inverse function \( f^{-1}(x) \), the domain is the range of the original function. Therefore, the domain of \( f^{-1}(x) \) is:
\[
(0, \infty)
\]

#### (ii) The range:
The domain of the original function \( f(x) = e^{2x + 1} \) is all real numbers, \( (-\infty, \infty) \). Therefore, the range of the inverse function \( f^{-1}(x) \) is also all real numbers:
\[
(-\infty, \infty)
\]

### Summary:
- \( f^{-1}(x) = \frac{\ln(x) - 1}{2} \)
- Domain of \( f^{-1}(x) \): \( (0, \infty) \)
- Range of \( f^{-1}(x) \): \( (-\infty, \infty) \)

Would you like further clarification on any of these steps?

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Here are some related questions you might find helpful:
1. How do you verify that a function and its inverse are correct?
2. What is the process for finding the inverse of other types of functions (e.g., trigonometric functions)?
3. How do logarithmic properties help in simplifying inverse functions?
4. What is the significance of the domain and range in inverse functions?
5. How do the graphs of a function and its inverse relate geometrically?

**Tip:** Always ensure the domain and range of the original function are considered when determining the domain and range of the inverse function!

Let f(x) = e^(2x + 1). (a) Determine f^(-1)(x). (b) For the function f^(-1)(x), determine (i) the domain and (ii) the range.

Learn how to find the inverse of the exponential function f(x) = e^(2x + 1) with detailed steps. The solution includes determining the domain and range of the inverse function, covering concepts of logarithms and exponential properties.

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