https://file.aiquickdraw.com/mathbot/file/17289524888630xfjppcq.jpg

Let's work through each of the functions in the image to find their inverses step by step.

### 1. \( g(x) = \frac{x - 5}{x + 4} \)

To find the inverse:
- Replace \( g(x) \) with \( y \):  
  \( y = \frac{x - 5}{x + 4} \)
- Swap \( x \) and \( y \) (since we're finding the inverse):  
  \( x = \frac{y - 5}{y + 4} \)
- Solve for \( y \):
  1. Multiply both sides by \( y + 4 \):  
     \( x(y + 4) = y - 5 \)
  2. Expand:  
     \( xy + 4x = y - 5 \)
  3. Group all terms involving \( y \) on one side:  
     \( xy - y = -4x - 5 \)
  4. Factor out \( y \) on the left:  
     \( y(x - 1) = -4x - 5 \)
  5. Divide both sides by \( x - 1 \):  
     \( y = \frac{-4x - 5}{x - 1} \)

So, the inverse is:  
\[
g^{-1}(x) = \frac{-4x - 5}{x - 1}
\]

---

### 2. \( f(x) = 4(x - 5) + 3 \)

To find the inverse:
- Replace \( f(x) \) with \( y \):  
  \( y = 4(x - 5) + 3 \)
- Swap \( x \) and \( y \):  
  \( x = 4(y - 5) + 3 \)
- Solve for \( y \):
  1. Subtract 3 from both sides:  
     \( x - 3 = 4(y - 5) \)
  2. Divide by 4:  
     \( \frac{x - 3}{4} = y - 5 \)
  3. Add 5 to both sides:  
     \( y = \frac{x - 3}{4} + 5 \)

So, the inverse is:  
\[
f^{-1}(x) = \frac{x - 3}{4} + 5
\]

---

### 3. \( f(x) = 6\sqrt{x - 9} \)

To find the inverse:
- Replace \( f(x) \) with \( y \):  
  \( y = 6\sqrt{x - 9} \)
- Swap \( x \) and \( y \):  
  \( x = 6\sqrt{y - 9} \)
- Solve for \( y \):
  1. Divide by 6:  
     \( \frac{x}{6} = \sqrt{y - 9} \)
  2. Square both sides:  
     \( \left(\frac{x}{6}\right)^2 = y - 9 \)
  3. Add 9 to both sides:  
     \( y = \left(\frac{x}{6}\right)^2 + 9 \)

So, the inverse is:  
\[
f^{-1}(x) = \left(\frac{x}{6}\right)^2 + 9
\]

---

### 4. \( h(x) = \frac{2x - 1}{x} \)

To find the inverse:
- Replace \( h(x) \) with \( y \):  
  \( y = \frac{2x - 1}{x} \)
- Swap \( x \) and \( y \):  
  \( x = \frac{2y - 1}{y} \)
- Solve for \( y \):
  1. Multiply both sides by \( y \):  
     \( xy = 2y - 1 \)
  2. Group all terms involving \( y \) on one side:  
     \( xy - 2y = -1 \)
  3. Factor out \( y \) on the left:  
     \( y(x - 2) = -1 \)
  4. Divide both sides by \( x - 2 \):  
     \( y = \frac{-1}{x - 2} \)

So, the inverse is:  
\[
h^{-1}(x) = \frac{-1}{x - 2}
\]

---

### 5. \( m(x) = x - 5 + 1 \)

This simplifies to:  
\( m(x) = x - 4 \)

To find the inverse:
- Replace \( m(x) \) with \( y \):  
  \( y = x - 4 \)
- Swap \( x \) and \( y \):  
  \( x = y - 4 \)
- Solve for \( y \):  
  \( y = x + 4 \)

So, the inverse is:  
\[
m^{-1}(x) = x + 4
\]

---

Would you like more details on any specific step or function?

### 5 Related Questions:
1. How do you verify if a function and its inverse are correct?
2. Can any function have an inverse?
3. How does the domain and range of a function relate to its inverse?
4. What are common types of functions that require extra steps to find the inverse?
5. Why is it important to understand inverses in algebra and calculus?

### Tip:  
To check if two functions are inverses of each other, compose them (\( f(f^{-1}(x)) \)) and see if the result is \( x \).

Give the inverse of the following functions: g(x) = (x - 5) / (x + 4), f(x) = 4(x - 5) + 3, f(x) = 6√(x - 9), h(x) = (2x - 1) / x, m(x) = x - 5 + 1.

Learn how to find the inverses of rational and radical functions, including g(x) = (x - 5) / (x + 4), f(x) = 6√(x - 9), and more. This guide is perfect for high school students studying algebra and precalculus.

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