To determine the domain and range of the inverse of the one-to-one function $f(x) = 2x - 7$, we need to follow these steps:

1. Find the Inverse Function:

To find the inverse of a function, we solve for $x$ in terms of $y$, where $f(x) = y$.

Start with: $y = 2x - 7$ Solve for $x$: $y + 7 = 2x$ $x = \frac{y + 7}{2}$ So, the inverse function $f^{-1}(x)$ is: $f^{-1}(x) = \frac{x + 7}{2}$

2. Domain of the Inverse Function:

The domain of the inverse function is the same as the range of the original function.

• The original function $f(x) = 2x - 7$ is a linear function, and linear functions have a range of all real numbers, i.e., $(-\infty, \infty)$.

Thus, the domain of the inverse function $f^{-1}(x)$ is: $(-\infty, \infty)$

3. Range of the Inverse Function:

The range of the inverse function is the same as the domain of the original function.

• The domain of $f(x) = 2x - 7$ is all real numbers, $(-\infty, \infty)$, because linear functions are defined for all real values of $x$.

Thus, the range of the inverse function $f^{-1}(x)$ is: $(-\infty, \infty)$

Conclusion:

• The domain of $f^{-1}(x) = \frac{x + 7}{2}$ is $(-\infty, \infty)$.
• The range of $f^{-1}(x) = \frac{x + 7}{2}$ is $(-\infty, \infty)$.

Would you like further details on how to verify or visualize this? Here are some related questions:

1. How can we graph the original function and its inverse on the same axes?
2. What happens to the domain and range if the function has restrictions?
3. How do we verify if a function is one-to-one?
4. How does the derivative help in identifying if a function has an inverse?
5. What is the importance of the horizontal line test in determining one-to-one functions?

Tip: Inverse functions reverse the roles of domain and range, so always swap these when dealing with inverse functions.