https://file.aiquickdraw.com/mathbot/file/1729271446271hv115o84.jpg

Let's work through the functions provided in the image one by one to determine if each has an inverse, and if it does, find the inverse function and any restrictions on the domain. We'll start with the first few and progress systematically.

### 13. \( g(x) = -3x^4 + 6x^2 - x \)
- **Does it have an inverse?** 
  This is a polynomial function of degree 4, meaning it is not one-to-one (a necessary condition for a function to have an inverse) because higher-degree polynomials like this are generally not invertible due to their multi-valued nature over parts of the domain.
  
  **Conclusion:** No inverse.

### 14. \( f(x) = \frac{4x^5 - 8x^4}{x} \)
- **Does it have an inverse?**
  First, simplify:
  \[
  f(x) = 4x^4 - 8x^3
  \]
  This is still a polynomial of degree 4, which means it's not one-to-one.

  **Conclusion:** No inverse.

### 15. \( h(x) = x^7 + 2x^3 - 10x^2 \)
- **Does it have an inverse?**
  This is a combination of terms with degrees 7, 3, and 2. Polynomials of this type are not generally one-to-one.

  **Conclusion:** No inverse.

### 16. \( f(x) = \sqrt{x + 8} \)
- **Does it have an inverse?**
  The square root function is one-to-one on its restricted domain \( x \geq -8 \), so it does have an inverse.
  
  - **Inverse:** To find the inverse, we swap \( x \) and \( y \) and solve for \( y \):
    \[
    y = \sqrt{x + 8} \quad \Rightarrow \quad x = \sqrt{y + 8}
    \quad \Rightarrow \quad y = x^2 - 8
    \]
  - **Domain Restriction:** \( x \geq -8 \) because the square root function requires non-negative values under the radical.

  **Conclusion:** Inverse is \( f^{-1}(x) = x^2 - 8 \), with domain \( x \geq -8 \).

### 17. \( f(x) = \sqrt{6 - x^2} \)
- **Does it have an inverse?**
  This is a square root function, but with a negative quadratic under the radical. To have real values, \( 6 - x^2 \geq 0 \), meaning \( x \in [-\sqrt{6}, \sqrt{6}] \). The function is one-to-one on this interval.
  
  - **Inverse:** Swapping \( x \) and \( y \),
    \[
    y = \sqrt{6 - x^2} \quad \Rightarrow \quad x = \sqrt{6 - y^2}
    \quad \Rightarrow \quad y = \sqrt{6 - x^2}
    \]
    The inverse would be:
    \[
    f^{-1}(x) = \pm \sqrt{6 - x^2}
    \]
    but only the positive branch would be valid since square roots are defined as positive.

  **Conclusion:** Inverse is \( f^{-1}(x) = \sqrt{6 - x^2} \), with domain \( x \in [-\sqrt{6}, \sqrt{6}] \).

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Would you like me to continue with more of the functions, or do you have specific ones in mind? 

Here are 5 follow-up questions:
1. Do you need help with any specific function in the list?
2. Would you like a detailed explanation of inverse functions?
3. Should we revisit the conditions for determining if a function has an inverse?
4. Would you like more practice finding inverses for functions with restrictions?
5. Are you comfortable with domain restrictions when finding inverses?

**Tip:** A function needs to be one-to-one (each \( y \) corresponds to one \( x \)) in order to have an inverse. You can check this by applying the horizontal line test!

Determine whether each function has an inverse function. If it does, find the inverse function and state any restriction on its domain.

This set of problems helps determine if a function has an inverse and how to find it. Explore inverse functions for polynomials, square root functions, and rational expressions with detailed domain restrictions. Suitable for Grades 10-12 students.

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