Find the Range of \(f(x)=\dfrac{|x-4|}{x-4}\)

Solution

Given

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

We consider two cases based on the sign of \(x-4\).

Case 1: \(x>4\)

Then,

\[ |x-4|=x-4 \]

Therefore,

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

Case 2: \(x<4\)

Then,

\[ |x-4|=-(x-4) \]

Therefore,

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

Restriction

At

\[ x=4 \]

denominator becomes zero, so the function is not defined.

Hence the function takes only two values:

\[ -1 \quad \text{and} \quad 1 \]