The range of the function `f(x)=(|x-4|)/(x-4)` is

Options

A. A. `[1,oo)`
B. B. `{-1,1}`
C. C. `R`
D. D. `{-1,3}`

Answer

To find the range of the function \( f(x) = \frac{|x-4|}{x-4} \), we will analyze the function based on the value of \( x \) relative to 4.

### Step 1: Determine the domain of the function
The function \( f(x) \) is defined for all real numbers except where the denominator is zero. Therefore, we need to find when \( x - 4 = 0 \).

\[ x - 4 = 0 \implies x = 4 \]

Thus, the domain of \( f(x) \) is all real numbers except \( x = 4 \):
\[ \text{Domain} = \mathbb{R} \setminus \{4\} \]

**Hint:** Check where the denominator becomes zero to determine the domain.

### Step 2: Analyze the function for \( x < 4 \)
When \( x < 4 \), the expression \( |x - 4| \) becomes \( -(x - 4) \) because \( x - 4 \) is negative. Therefore, we can rewrite the function as:

\[ f(x) = \frac{|x - 4|}{x - 4} = \frac{-(x - 4)}{x - 4} = -1 \]

So, for all \( x < 4 \), \( f(x) = -1 \).

**Hint:** Remember that the absolute value function changes the sign based on whether the input is positive or negative.

### Step 3: Analyze the function for \( x > 4 \)
When \( x > 4 \), the expression \( |x - 4| \) becomes \( x - 4 \) because \( x - 4 \) is positive. Therefore, we can rewrite the function as:

\[ f(x) = \frac{|x - 4|}{x - 4} = \frac{x - 4}{x - 4} = 1 \]

So, for all \( x > 4 \), \( f(x) = 1 \).

**Hint:** For values greater than 4, the absolute value does not change the sign of the expression.

### Step 4: Combine the results
From our analysis:
- For \( x < 4 \), \( f(x) = -1 \)
- For \( x > 4 \), \( f(x) = 1 \)

Since \( f(x) \) is not defined at \( x = 4 \), the function takes on the values -1 and 1, but does not include these points.

### Conclusion: Determine the range
The range of the function \( f(x) \) is therefore:
\[ \text{Range} = \{-1, 1\} \]

Thus, the correct answer is that the range of the function is \(-1\) and \(1\).

**Final Answer:** The range of the function \( f(x) = \frac{|x-4|}{x-4} \) is \(-1\) and \(1\).