Calculate the inverse of the function. If the inverse does not exist, explain why. f(x) = 6 + 5/x

To determine if the inverse of the function \( f(x) = 6 + \frac{5}{x} \) exists and, if so, to find it, we follow these steps:

### Step 1: Check if the function is one-to-one
A function must be one-to-one (injective) for an inverse to exist. To check if \( f(x) \) is one-to-one, let's analyze the behavior of the function:

\[
f(x) = 6 + \frac{5}{x}
\]

- The term \( \frac{5}{x} \) is strictly decreasing for \( x \neq 0 \).
- Adding 6 to this does not change the fact that the function will still be strictly decreasing, since the sum of a decreasing function and a constant is still decreasing.

Since the function is strictly decreasing, it passes the horizontal line test, meaning it is one-to-one, and therefore an inverse exists.

### Step 2: Solve for the inverse
To find the inverse, we follow the standard process:

1. Replace \( f(x) \) with \( y \):
   \[
   y = 6 + \frac{5}{x}
   \]

2. Swap \( y \) and \( x \) to reflect the relationship for the inverse:
   \[
   x = 6 + \frac{5}{y}
   \]

3. Solve for \( y \) in terms of \( x \):

   - Subtract 6 from both sides:
     \[
     x - 6 = \frac{5}{y}
     \]

   - Multiply both sides by \( y \) to eliminate the fraction:
     \[
     y(x - 6) = 5
     \]

   - Divide by \( x - 6 \):
     \[
     y = \frac{5}{x - 6}
     \]

### Step 3: Conclusion
Thus, the inverse function is:

\[
f^{-1}(x) = \frac{5}{x - 6}
\]

### Domain Consideration
The function \( f(x) = 6 + \frac{5}{x} \) has a restriction that \( x \neq 0 \) (since division by zero is undefined). Similarly, the inverse function \( f^{-1}(x) = \frac{5}{x - 6} \) is undefined when \( x = 6 \), so \( x \neq 6 \).

Let me know if you want more details, or if you have any questions.

Here are 5 related questions:
1. What is the domain of the inverse function \( f^{-1}(x) \)?
2. How do you verify that \( f(f^{-1}(x)) = x \)?
3. What is the graphical interpretation of a function's inverse?
4. Can all functions have inverses? Why or why not?
5. How do you determine if a function is one-to-one?

**Tip:** A function is invertible if it is both one-to-one and onto (bijective). Always check injectivity before finding the inverse.

Learn how to calculate the inverse of the function f(x) = 6 + 5/x, step by step. This solution covers algebraic concepts such as one-to-one functions, the horizontal line test, and domain restrictions. Suitable for students in Grades 9-12.

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