https://file.aiquickdraw.com/mathbot/file/1726620902065ro2b40mi.jpg

Let's solve the problem of finding the inverse of the three functions given in the image one by one:

### 1. Find the inverse of \( f(x) = 4x + 3 \)

#### Steps:
1. Replace \( f(x) \) with \( y \): 
   \[
   y = 4x + 3
   \]
2. Swap \( x \) and \( y \):
   \[
   x = 4y + 3
   \]
3. Solve for \( y \):
   \[
   x - 3 = 4y
   \]
   \[
   y = \frac{x - 3}{4}
   \]
Thus, the inverse is:
\[
f^{-1}(x) = \frac{x - 3}{4}
\]

### 2. Find the inverse of \( f(x) = \frac{1}{5x - 6} \)

#### Steps:
1. Replace \( f(x) \) with \( y \):
   \[
   y = \frac{1}{5x - 6}
   \]
2. Swap \( x \) and \( y \):
   \[
   x = \frac{1}{5y - 6}
   \]
3. Solve for \( y \):
   \[
   x(5y - 6) = 1
   \]
   \[
   5xy - 6x = 1
   \]
   \[
   5xy = 6x + 1
   \]
   \[
   y = \frac{6x + 1}{5x}
   \]
Thus, the inverse is:
\[
f^{-1}(x) = \frac{6x + 1}{5x}
\]

### 3. Find the inverse of \( f(x) = \frac{x - 7}{3x + 1} \)

#### Steps:
1. Replace \( f(x) \) with \( y \):
   \[
   y = \frac{x - 7}{3x + 1}
   \]
2. Swap \( x \) and \( y \):
   \[
   x = \frac{y - 7}{3y + 1}
   \]
3. Solve for \( y \):
   \[
   x(3y + 1) = y - 7
   \]
   \[
   3xy + x = y - 7
   \]
   \[
   3xy - y = -x - 7
   \]
   \[
   y(3x - 1) = -x - 7
   \]
   \[
   y = \frac{-x - 7}{3x - 1}
   \]
Thus, the inverse is:
\[
f^{-1}(x) = \frac{-x - 7}{3x - 1}
\]

---

Would you like further clarification on any of these steps? Here are some related questions you might find useful:

1. What are the general steps to find the inverse of any function?
2. How can you verify if the computed inverse is correct?
3. What are the conditions for a function to have an inverse?
4. How do domain and range change when finding an inverse?
5. Can every function have an inverse? Why or why not?

**Tip:** To check if the inverse is correct, composing the function and its inverse should return \( x \). That is, \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

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

Explore the step-by-step process of finding the inverses for linear and rational functions such as f(x) = 4x + 3, f(x) = 1 / (5x - 6), and f(x) = (x - 7) / (3x + 1). This detailed guide includes algebraic manipulations and the use of the function inversion theorem, suitable for college-level students.

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