find the inverse of the following. 1. f(x)=1/4x-2 and f(x)=2x^3-5 and f(x)3/x-2

To find the inverse of a function \( f(x) \), you typically swap \( x \) and \( y \) (where \( y = f(x) \)) and then solve for \( y \). Here are the steps for each given function:

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

**Steps to find the inverse:**

1. Let \( y = \frac{1}{4x} - 2 \).
2. Swap \( x \) and \( y \): \( x = \frac{1}{4y} - 2 \).
3. Solve for \( y \):

   \[
   x + 2 = \frac{1}{4y}
   \]

   Multiply both sides by \( 4y \):

   \[
   4y(x + 2) = 1
   \]

   Divide by \( x + 2 \):

   \[
   y = \frac{1}{4(x + 2)}
   \]

So, the inverse is:

\[
f^{-1}(x) = \frac{1}{4(x + 2)}
\]

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

**Steps to find the inverse:**

1. Let \( y = 2x^3 - 5 \).
2. Swap \( x \) and \( y \): \( x = 2y^3 - 5 \).
3. Solve for \( y \):

   \[
   x + 5 = 2y^3
   \]

   Divide by 2:

   \[
   y^3 = \frac{x + 5}{2}
   \]

   Take the cube root:

   \[
   y = \sqrt[3]{\frac{x + 5}{2}}
   \]

So, the inverse is:

\[
f^{-1}(x) = \sqrt[3]{\frac{x + 5}{2}}
\]

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

**Steps to find the inverse:**

1. Let \( y = \frac{3}{x - 2} \).
2. Swap \( x \) and \( y \): \( x = \frac{3}{y - 2} \).
3. Solve for \( y \):

   \[
   y - 2 = \frac{3}{x}
   \]

   Add 2 to both sides:

   \[
   y = \frac{3}{x} + 2
   \]

So, the inverse is:

\[
f^{-1}(x) = \frac{3}{x} + 2
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you verify if a function and its inverse are correct?
2. What is the graphical interpretation of a function and its inverse?
3. How do you find the inverse of a function that is not one-to-one?
4. Can all functions have inverses? Why or why not?
5. How do you determine if a function is invertible?

**Tip:** To check if your inverse is correct, substitute the inverse function into the original function and see if it simplifies to \( x \).

Explore how to find the inverse of functions with detailed examples and step-by-step solutions. This page covers functions like 1. f(x) = 1/(4x) - 2, 2. f(x) = 2x^3 - 5, and 3. f(x) = 3/(x - 2). Suitable for advanced high school students.

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