Let's work through the problem in the image for $f(x) = 3x + 5$.

(a) Domain of $f$:

The function $f(x) = 3x + 5$ is a linear function, which means it is defined for all real values of $x$. Therefore, the domain of $f$ is: $\text{Domain of } f = (-\infty, \infty)$

(b) Range of $f$:

Since $f(x) = 3x + 5$ is a linear function with no restrictions, the range is also all real numbers. As $x$ takes on any real value, $f(x)$ covers all real numbers. Hence, the range of $f$ is: $\text{Range of } f = (-\infty, \infty)$

(c) Formula for $f^{-1}$ (Inverse of $f$):

To find the inverse, we follow these steps:

1. Start with the equation: $y = 3x + 5$.
2. Solve for $x$: $y - 5 = 3x$ $x = \frac{y - 5}{3}$
3. Replace $y$ with $x$ to get the inverse function: $f^{-1}(x) = \frac{x - 5}{3}$

(d) Domain of $f^{-1}$:

Since the domain of the inverse function is the range of the original function, and we found that the range of $f(x)$ is all real numbers, the domain of $f^{-1}(x)$ is: $\text{Domain of } f^{-1} = (-\infty, \infty)$

(e) Range of $f^{-1}$:

The range of $f^{-1}$ is the same as the domain of the original function $f(x)$, which is also all real numbers. Hence, the range of $f^{-1}(x)$ is: $\text{Range of } f^{-1} = (-\infty, \infty)$

Would you like further explanation on any of these parts? Let me know if you have any questions!

Here are 5 related questions for further practice:

1. What is the inverse of the function $f(x) = 2x - 3$?
2. How do you find the inverse of a quadratic function?
3. Can all functions have inverses? Why or why not?
4. What is the significance of the range and domain in real-world problems?
5. How does a horizontal line test determine if a function is invertible?

Tip: Always remember that a function must be one-to-one (i.e., it passes the horizontal line test) in order to have an inverse.