Range of `f(x) = (1)/(2x-1)` is

Options

A. `(-oo, oo)`
B. `(-oo, 0]`
C. `[0, oo)`
D. R - {0}

Answer

To find the range of the function \( f(x) = \frac{1}{2x - 1} \), we will follow these steps:

### Step 1: Replace \( f(x) \) with \( y \)
We start by setting \( y = f(x) \):
\[ y = \frac{1}{2x - 1} \]

### Step 2: Rearrange the equation
Next, we can rearrange this equation to express \( x \) in terms of \( y \):
\[ y(2x - 1) = 1 \]
Expanding this gives:
\[ 2xy - y = 1 \]

### Step 3: Solve for \( x \)
Now, we isolate \( x \):
\[ 2xy = 1 + y \]
\[ x = \frac{1 + y}{2y} \]

### Step 4: Determine the domain of the new function
The expression we derived for \( x \) is valid as long as the denominator is not zero. Therefore, we need to find when \( 2y \neq 0 \):
\[ 2y \neq 0 \implies y \neq 0 \]

### Step 5: Conclude the range of \( f(x) \)
Since \( y \) can take any real number except for 0, we conclude that the range of the function \( f(x) = \frac{1}{2x - 1} \) is:
\[ \text{Range} = \mathbb{R} \setminus \{0\} \]

### Final Answer
The range of \( f(x) = \frac{1}{2x - 1} \) is all real numbers except 0.

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