Inverse of `-3+(5)/(x-2)` is

Options

A. `(2x+11)/(x+3)`
B. `-3+ (x-2)/(5)`
C. `-(1)/(3) + (x-2)/(5)`
D. `3-(5)/(x-2)`

Answer

To find the inverse of the function \( f(x) = -3 + \frac{5}{x - 2} \), we will follow these steps:

### Step 1: Rewrite the function
Start with the function:
\[ f(x) = -3 + \frac{5}{x - 2} \]
We can rewrite this as:
\[ f(x) + 3 = \frac{5}{x - 2} \]

**Hint:** Isolate the fraction by adding 3 to both sides.

### Step 2: Cross-multiply
Now, we can cross-multiply to eliminate the fraction:
\[ (f(x) + 3)(x - 2) = 5 \]

**Hint:** Remember that cross-multiplying involves multiplying both sides of the equation by the denominator.

### Step 3: Expand the equation
Expanding the left side gives:
\[ f(x)x - 2f(x) + 3x - 6 = 5 \]

**Hint:** Distribute the terms on the left side carefully.

### Step 4: Rearrange the equation
Rearranging the equation to isolate terms involving \( x \):
\[ f(x)x - 2f(x) + 3x - 11 = 0 \]

**Hint:** Move all terms to one side to set the equation to zero.

### Step 5: Solve for \( x \)
Now, we need to express \( x \) in terms of \( f(x) \):
\[ x(f(x) + 3) = 11 + 2f(x) \]
\[ x = \frac{11 + 2f(x)}{f(x) + 3} \]

**Hint:** Factor out \( x \) to isolate it on one side.

### Step 6: Replace \( f(x) \) with \( x \)
To find the inverse function, replace \( f(x) \) with \( x \):
\[ f^{-1}(x) = \frac{11 + 2x}{x + 3} \]

**Hint:** This substitution is key to finding the inverse function.

### Final Result
Thus, the inverse of the function \( f(x) = -3 + \frac{5}{x - 2} \) is:
\[ f^{-1}(x) = \frac{2x + 11}{x + 3} \]