If `f(x)=(|x-2||x-1|)/(|x-3|)`, then value of `f(-2)` is

Options

A. `(12)/(5)`
B. `(-12)/(5)`
C. 12
D. 5

Answer

To find the value of \( f(-2) \) for the function \( f(x) = \frac{|x-2||x-1|}{|x-3|} \), we will substitute \( -2 \) into the function and simplify step by step.

### Step 1: Substitute \( -2 \) into the function
We start with the function:
\[ f(x) = \frac{|x-2||x-1|}{|x-3|} \]
Now, we substitute \( x = -2 \):
\[ f(-2) = \frac{|-2-2||-2-1|}{|-2-3|} \]

### Step 2: Simplify the expressions inside the absolute values
Now, we calculate each term inside the absolute values:
- For \( |-2-2| \):
\[ |-2-2| = |-4| = 4 \]
- For \( |-2-1| \):
\[ |-2-1| = |-3| = 3 \]
- For \( |-2-3| \):
\[ |-2-3| = |-5| = 5 \]

### Step 3: Substitute the simplified values back into the function
Now we can substitute these values back into the function:
\[ f(-2) = \frac{4 \cdot 3}{5} \]

### Step 4: Calculate the final value
Now we perform the multiplication and division:
\[ f(-2) = \frac{12}{5} \]

Thus, the value of \( f(-2) \) is:
\[ \boxed{\frac{12}{5}} \]