What is the range of the function f(x)=`3/(x-5)-1`?

Options

A. All real numbers
B. All real numbers except 5
C. All real numbers except 0
D. All real numbers except-1

Explanation

To find the range of the function \( f(x) = \frac{3}{x-5} - 1 \), we can follow these steps:

### Step 1: Rewrite the function
Let \( y = f(x) \). Then, we have:
\[ y = \frac{3}{x-5} - 1 \]

### Step 2: Solve for \( x \)
Rearranging the equation to solve for \( x \):
\[ y + 1 = \frac{3}{x-5} \]
Now, multiply both sides by \( x - 5 \) (assuming \( x \neq 5 \)):
\[ (y + 1)(x - 5) = 3 \]
Expanding this gives:
\[ yx - 5y + x - 5 = 3 \]
Rearranging terms:
\[ yx + x = 3 + 5y + 5 \]
Factoring out \( x \):
\[ x(y + 1) = 8 + 5y \]
Finally, solving for \( x \):
\[ x = \frac{8 + 5y}{y + 1} \]

### Step 3: Determine the domain of \( f^{-1}(y) \)
The expression for \( x \) is defined for all \( y \) except when the denominator is zero:
\[ y + 1 \neq 0 \implies y \neq -1 \]

### Step 4: Conclusion about the range of \( f(x) \)
Since the only restriction on \( y \) is that \( y \neq -1 \), the range of the function \( f(x) \) is:
\[ \text{Range of } f(x) = \mathbb{R} \setminus \{-1\} \]

### Final Answer
The range of the function \( f(x) = \frac{3}{x-5} - 1 \) is all real numbers except -1.

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