https://file.aiquickdraw.com/mathbot/file/17285197790445vaspkuo.jpg

Let's go through the functions in the image and find their inverses.

### 1. \( f(x) = \frac{6}{x} + 7 \)

To find the inverse, follow these steps:

- Step 1: Replace \( f(x) \) with \( y \):
  \[
  y = \frac{6}{x} + 7
  \]
- Step 2: Swap \( x \) and \( y \) to start solving for the inverse:
  \[
  x = \frac{6}{y} + 7
  \]
- Step 3: Isolate \( y \):
  \[
  x - 7 = \frac{6}{y}
  \]
  \[
  y = \frac{6}{x - 7}
  \]
Thus, the inverse is:
\[
f^{-1}(x) = \frac{6}{x - 7}
\]

---

### 2. \( f(x) = 4(x - 5) + 7 \)

Steps:

- Step 1: Replace \( f(x) \) with \( y \):
  \[
  y = 4(x - 5) + 7
  \]
- Step 2: Expand and simplify:
  \[
  y = 4x - 20 + 7 = 4x - 13
  \]
- Step 3: Swap \( x \) and \( y \):
  \[
  x = 4y - 13
  \]
- Step 4: Solve for \( y \):
  \[
  4y = x + 13
  \]
  \[
  y = \frac{x + 13}{4}
  \]
Thus, the inverse is:
\[
f^{-1}(x) = \frac{x + 13}{4}
\]

---

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

Steps:

- Step 1: Replace \( f(x) \) with \( y \):
  \[
  y = 3 \sqrt{x - 3}
  \]
- Step 2: Swap \( x \) and \( y \):
  \[
  x = 3 \sqrt{y - 3}
  \]
- Step 3: Solve for \( y \):
  \[
  \frac{x}{3} = \sqrt{y - 3}
  \]
  \[
  \left(\frac{x}{3}\right)^2 = y - 3
  \]
  \[
  y = \frac{x^2}{9} + 3
  \]
Thus, the inverse is:
\[
f^{-1}(x) = \frac{x^2}{9} + 3
\]

---

### 4. \( h(x) = \frac{2x + 6}{x} \)

Steps:

- Step 1: Simplify the function:
  \[
  h(x) = 2 + \frac{6}{x}
  \]
- Step 2: Replace \( h(x) \) with \( y \):
  \[
  y = 2 + \frac{6}{x}
  \]
- Step 3: Swap \( x \) and \( y \):
  \[
  x = 2 + \frac{6}{y}
  \]
- Step 4: Solve for \( y \):
  \[
  x - 2 = \frac{6}{y}
  \]
  \[
  y = \frac{6}{x - 2}
  \]
Thus, the inverse is:
\[
h^{-1}(x) = \frac{6}{x - 2}
\]

---

### 5. \( m(x) = 2(x - 1)^2 + 3 \)

Steps:

- Step 1: Replace \( m(x) \) with \( y \):
  \[
  y = 2(x - 1)^2 + 3
  \]
- Step 2: Swap \( x \) and \( y \):
  \[
  x = 2(y - 1)^2 + 3
  \]
- Step 3: Solve for \( y \) (this requires multiple steps):

1. Subtract 3 from both sides:
   \[
   x - 3 = 2(y - 1)^2
   \]
2. Divide by 2:
   \[
   \frac{x - 3}{2} = (y - 1)^2
   \]
3. Take the square root of both sides:
   \[
   y - 1 = \pm \sqrt{\frac{x - 3}{2}}
   \]
4. Add 1 to isolate \( y \):
   \[
   y = 1 \pm \sqrt{\frac{x - 3}{2}}
   \]
Thus, the inverse is:
\[
m^{-1}(x) = 1 \pm \sqrt{\frac{x - 3}{2}}
\]

---

Would you like more detailed steps for any of these inverses or have any questions?

### Five related questions:
1. What are the conditions for a function to have an inverse?
2. Can you graph one of the original functions and its inverse?
3. How do you verify that two functions are inverses of each other?
4. How does the domain of a function affect the inverse?
5. What happens to the range of a function when you find its inverse?

**Tip:** When solving for an inverse function, always remember to check if the function is one-to-one (bijective) to ensure an inverse exists!

Find the inverse of the following functions: 1) f(x) = 6/x + 7 2) f(x) = 4(x - 5) + 7 3) f(x) = 3√(x - 3) 4) h(x) = (2x + 6)/x 5) m(x) = 2(x - 1)^2 + 3

Step-by-step solutions to find the inverse of various functions, including rational functions, quadratic forms, and square roots. This guide helps high school students understand the process of solving inverse functions with detailed examples. Suitable for Grades 10-12.

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