Evaluate \(f(g(x))\)

🎥 Video Explanation

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📝 Question

Given:

\[ g(x)=1+x-[x] \]

\[ f(x)= \begin{cases} -1, & x<0 \\ 0, & x=0 \\ 1, & x>0 \end{cases} \]

• (a) \(x\)
• (b) \(1\)
• (c) \(f(x)\)
• (d) \(g(x)\)

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✅ Solution

🔹 Step 1: Understand \(g(x)\)

\[ x-[x] \in [0,1) \]

So:

\[ g(x)=1+(x-[x]) \in [1,2) \] —

🔹 Step 2: Apply \(f\)

Since \(g(x) \in [1,2)\), it is always:

\[ g(x) > 0 \]

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🔹 Step 3: Use Definition of \(f\)

For all positive inputs:

\[ f(g(x)) = 1 \] —

🔹 Final Answer

\[ \boxed{\text{Option (b): } 1} \]