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

📝 Question

Let:

\[ A=\{x\in \mathbb{R}:-4\le x\le 4,\ x\ne 0\} \]

\[ f:A\to \mathbb{R}, \quad f(x)=\frac{|x|}{x} \]

Find the range of \(f\).

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✅ Solution

🔹 Step 1: Consider cases

Case 1: \(x>0\)

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

Case 2: \(x<0\)

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

🔹 Step 2: Combine values

The function takes only two values:

:contentReference[oaicite:0]{index=0} —

🎯 Final Answer

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

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🚀 Exam Shortcut

• \(\frac{|x|}{x} = 1\) if \(x>0\)
• \(\frac{|x|}{x} = -1\) if \(x<0\)
• \(x=0\) not defined
• Range = \{-1, 1\}