Find Range of \(f(x)=\frac{|x-1|}{x-1}\)

📝 Question

Find the range of the function:

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

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✅ Solution

🔹 Step 1: Domain restriction

Denominator cannot be zero:

\[ x-1 \ne 0 \Rightarrow x \ne 1 \] —

🔹 Step 2: Consider cases

Case 1: \(x>1\)

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

Case 2: \(x<1\)

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

🔹 Step 3: Combine values

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🎯 Final Answer

\[ \boxed{\{-1,\,1\}} \]

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🚀 Exam Shortcut

• Split into cases based on sign of \(x-1\)
• Expression becomes ±1
• Exclude \(x=1\)
• Range = \{-1, 1\}