https://file.aiquickdraw.com/mathbot/file/1726159130358k2y585q9.jpg

We are given the function \( f(x) = \frac{3}{x} \), which is one-to-one. Let's go step by step.

### Part (a): Find the inverse function \( f^{-1}(x) \)

To find the inverse of \( f(x) = \frac{3}{x} \), follow these steps:
1. Replace \( f(x) \) with \( y \), so we have:
   \[
   y = \frac{3}{x}
   \]
2. Swap \( x \) and \( y \), giving:
   \[
   x = \frac{3}{y}
   \]
3. Solve for \( y \):
   \[
   y = \frac{3}{x}
   \]
Therefore, the inverse function is:
\[
f^{-1}(x) = \frac{3}{x}
\]

### Part (b): Verify the inverse
We need to verify that:
1. \( f(f^{-1}(x)) = x \)
2. \( f^{-1}(f(x)) = x \)

Let’s check both:

1. \( f(f^{-1}(x)) \):
   \[
   f(f^{-1}(x)) = f\left( \frac{3}{x} \right) = \frac{3}{\frac{3}{x}} = x
   \]
   This holds true.

2. \( f^{-1}(f(x)) \):
   \[
   f^{-1}(f(x)) = f^{-1}\left( \frac{3}{x} \right) = \frac{3}{\frac{3}{x}} = x
   \]
   This also holds true.

Thus, the inverse is correctly verified.

### Part (a) - Choice
From the options in the image:
- \( f^{-1}(x) = \frac{3}{x} \)
  
Thus, the correct answer corresponds to **Option D**: \( f^{-1}(x) = \frac{3}{x} \), for \( x \neq 0 \).

---

Would you like more details on any part of this? Here are some related questions:

1. How do we verify if a function is one-to-one?
2. Why is it necessary to restrict the domain of the inverse function?
3. Can you explain why \( f(x) \) and \( f^{-1}(x) \) are reflections across the line \( y = x \)?
4. How do the concepts of domain and range apply to inverse functions?
5. How do you graph a function and its inverse?

**Tip:** When finding an inverse function, always remember to swap \( x \) and \( y \) and then solve for \( y \).

Find the inverse function for f(x) = 3/x and verify the correctness by showing f(f^-1(x)) = x and f^-1(f(x)) = x.

Learn how to find the inverse of the function f(x) = 3/x and verify it by solving for f(f^-1(x)) = x and f^-1(f(x)) = x. Suitable for high school students in grades 10-12, this problem focuses on inverse functions and algebraic manipulation.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation