What is the inverse function of f(x) = x/3-5? (Use g(x) to represent the inverse function)

To find the inverse function \( g(x) \) of the given function \( f(x) = \frac{x}{3} - 5 \), we follow these steps:

1. **Replace \( f(x) \) with \( y \):**
   \[
   y = \frac{x}{3} - 5
   \]
   
2. **Switch \( x \) and \( y \):**  
   To find the inverse, swap \( x \) and \( y \):
   \[
   x = \frac{y}{3} - 5
   \]
   
3. **Solve for \( y \):**  
   Now, solve for \( y \) in terms of \( x \).
   \[
   x + 5 = \frac{y}{3}
   \]
   Multiply both sides by 3:
   \[
   3(x + 5) = y
   \]
   So,
   \[
   y = 3(x + 5)
   \]
   
4. **Write the inverse function:**  
   Since \( y = g(x) \), we have:
   \[
   g(x) = 3(x + 5)
   \]

Thus, the inverse function is:
\[
g(x) = 3(x + 5)
\]

Would you like further details or have any questions?

Here are 5 related questions to expand on this topic:
1. How do you verify that two functions are inverses of each other?
2. What is the domain and range of the function \( f(x) = \frac{x}{3} - 5 \) and its inverse?
3. How would the graph of the function \( f(x) \) and its inverse \( g(x) \) look like?
4. What happens to the inverse if we modify \( f(x) \) to \( f(x) = \frac{x}{3} + 5 \)?
5. Can a function have more than one inverse? Why or why not?

**Tip:** The graph of a function and its inverse are symmetrical across the line \( y = x \).

What is the inverse function of f(x) = x/3 - 5? (Use g(x) to represent the inverse function)

Learn how to find the inverse function of f(x) = x/3 - 5. This detailed solution walks through the steps of switching variables and solving for y to determine the inverse function. Perfect for high school algebra students (Grades 9-12).

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