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get the inverse function of the following. give me the solution and answers

To solve the problem of finding the inverse functions, let's go through each function step by step.

### Problem 1: \( f(x) = \frac{5}{4}x - 3 \)

To find the inverse, follow these steps:

1. Replace \( f(x) \) with \( y \):
   \[
   y = \frac{5}{4}x - 3
   \]
   
2. Swap \( x \) and \( y \) to find the inverse:
   \[
   x = \frac{5}{4}y - 3
   \]
   
3. Solve for \( y \):
   \[
   x + 3 = \frac{5}{4}y
   \]
   \[
   y = \frac{4}{5}(x + 3)
   \]

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

### Problem 2: \( f(x) = \frac{3}{x - 2} \)

For this function:

1. Replace \( f(x) \) with \( y \):
   \[
   y = \frac{3}{x - 2}
   \]
   
2. Swap \( x \) and \( y \):
   \[
   x = \frac{3}{y - 2}
   \]
   
3. Solve for \( y \):
   \[
   x(y - 2) = 3
   \]
   \[
   xy - 2x = 3
   \]
   \[
   xy = 3 + 2x
   \]
   \[
   y = \frac{3 + 2x}{x}
   \]

So, the inverse function is:
\[
f^{-1}(x) = \frac{3 + 2x}{x}
\]

### Problem 3: \( f(x) = \frac{2x + 3}{x - 7} \)

Finally, for this function:

1. Replace \( f(x) \) with \( y \):
   \[
   y = \frac{2x + 3}{x - 7}
   \]
   
2. Swap \( x \) and \( y \):
   \[
   x = \frac{2y + 3}{y - 7}
   \]
   
3. Solve for \( y \):
   \[
   x(y - 7) = 2y + 3
   \]
   \[
   xy - 7x = 2y + 3
   \]
   \[
   xy - 2y = 7x + 3
   \]
   \[
   y(x - 2) = 7x + 3
   \]
   \[
   y = \frac{7x + 3}{x - 2}
   \]

So, the inverse function is:
\[
f^{-1}(x) = \frac{7x + 3}{x - 2}
\]

### Summary of Inverse Functions:
1. \( f^{-1}(x) = \frac{4}{5}(x + 3) \)
2. \( f^{-1}(x) = \frac{3 + 2x}{x} \)
3. \( f^{-1}(x) = \frac{7x + 3}{x - 2} \)

Would you like more details on any step or have any questions?

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Here are some related questions to deepen your understanding:
1. How can you verify that the function found is indeed the inverse?
2. What happens if the function is not one-to-one? Can it have an inverse?
3. How would you approach finding the inverse of a quadratic function?
4. Can you derive the inverse function for a trigonometric function, such as \( \sin(x) \)?
5. How does the domain and range of a function change when finding its inverse?

**Tip:** Always remember that not all functions have an inverse. For a function to have an inverse, it must be one-to-one (bijective), meaning each output is produced by exactly one input.

Learn how to find the inverse function for given functions f(x) = (5/4)x - 3, f(x) = 3/(x - 2), and f(x) = (2x + 3)/(x - 7). This page provides step-by-step solutions and inverse function formulas, covering properties of inverse functions. Suitable for Grades 10-12 students.

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