https://file.aiquickdraw.com/mathbot/file/1727894856784cihomcon.jpg

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:
\[
\text{Range of } f^{-1} = (-\infty, \infty)
\]

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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.

For the function f(x) = 3x + 5, find the domain, range, and inverse along with their respective domain and range.

Learn how to determine the domain, range, and inverse of the linear function f(x) = 3x + 5. The solution includes finding the inverse formula and understanding the domains and ranges of both the function and its inverse. Suitable for high school students in Grades 10-12.

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