https://file.aiquickdraw.com/mathbot/file/1726497336424x3te2u35.jpg

The image presents the inverse function \( f^{-1}(x) \) for the function \( f(x) = 3x + 3 \), as well as an attempt to verify that the inverse satisfies the property \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

### Correct Inverse Function:
The inverse function is correctly found as:
\[
f^{-1}(x) = \frac{x}{3} - 1
\]
This is checked as correct with a green tick in the image.

### Verification Process:
To verify the properties of the inverse, we compute two things:
1. **\( f(f^{-1}(x)) = x \)**:
    - The attempt shown mistakenly inputs \( 3x \) into \( f \), which is incorrect. The correct substitution should be \( f(f^{-1}(x)) = f\left( \frac{x}{3} - 1 \right) \), and then simplify.

2. **\( f^{-1}(f(x)) = x \)**:
    - Similarly, the verification process for \( f^{-1}(f(x)) \) mistakenly uses incorrect values and simplifications. The correct process is to substitute \( f(x) = 3x + 3 \) into \( f^{-1} \).

Would you like me to work through the corrections for these verifications in detail? Let me know!

### Follow-up Questions:
1. How do you find the inverse of a linear function?
2. What properties must an inverse function satisfy?
3. Why did the mistakes in the verification happen here?
4. How can you graphically verify if two functions are inverses?
5. Can every function have an inverse? Why or why not?

### Tip:
Always double-check the substitutions when verifying inverse functions to ensure the process is accurate!

Find the inverse function of f(x) = 3x + 3 and verify that f(f^-1(x)) = x and f^-1(f(x)) = x.

Explore how to find the inverse of the linear function f(x) = 3x + 3 and verify the properties f(f^-1(x)) = x and f^-1(f(x)) = x. This detailed solution covers key algebraic concepts and the step-by-step verification process. Suitable for Grades 9-12.

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