Range of `f(x) = (|x-5|)/(x-5)` is

Options

A. [-5, 5]
B. {-5, 5}
C. {-1,1}
D. {1,5}

Answer

To find the range of the function \( f(x) = \frac{|x-5|}{x-5} \), we will analyze the function based on the definition of the absolute value.

### Step 1: Understand the absolute value function
The absolute value function \( |x-5| \) can be defined piecewise:
- \( |x-5| = x-5 \) when \( x \geq 5 \)
- \( |x-5| = -(x-5) = 5-x \) when \( x < 5 \)

### Step 2: Define the function based on the piecewise definition
Using the piecewise definition of the absolute value, we can rewrite \( f(x) \) as:
- For \( x \geq 5 \):
\[ f(x) = \frac{x-5}{x-5} = 1 \]
- For \( x < 5 \):
\[ f(x) = \frac{5-x}{x-5} = \frac{-(x-5)}{x-5} = -1 \]

### Step 3: Determine the values of \( f(x) \)
From the piecewise definitions:
- When \( x \geq 5 \), \( f(x) = 1 \)
- When \( x < 5 \), \( f(x) = -1 \)

### Step 4: Identify the range of \( f(x) \)
The function takes on two distinct values:
- \( f(x) = 1 \) for all \( x \) greater than or equal to 5
- \( f(x) = -1 \) for all \( x \) less than 5

Thus, the range of \( f(x) \) is:
\[ \text{Range} = \{-1, 1\} \]

### Final Answer
The range of \( f(x) = \frac{|x-5|}{x-5} \) is \( \{-1, 1\} \).

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