Find \(f \circ f\) for Given Piecewise Function

📺 Video Explanation

📝 Question

Let:

\[ f(x)= \begin{cases} 1+x, & 0\le x\le 2 \\[4pt] 3-x, & 2

Find:

\[ f\circ f \]

---

✅ Solution

🔹 Step 1: Write composite function

By definition:

\[ (f\circ f)(x)=f(f(x)) \]

We first find range of \(f(x)\).

---

🔹 Step 2: Range of \(f(x)\)

• For \(0\le x\le2\): \(f(x)=1+x\Rightarrow 1\le f(x)\le3\)
• For \(2

So overall:

\[ 0\le f(x)\le3 \]

Thus, composition is defined on:

\[ 0\le x\le3 \]

---

🔹 Step 3: Case I: \(0\le x\le2\)

Here:

\[ f(x)=1+x \]

So:

\[ (f\circ f)(x)=f(1+x) \]

Now split again:

• If \(0\le x\le1\)

Then:

\[ 1+x\le2 \]

Use first rule:

\[ f(1+x)=1+(1+x)=x+2 \]

• If \(1

Then:

\[ 2<1+x\le3 \]

Use second rule:

\[ f(1+x)=3-(1+x)=2-x \]

---

🔹 Step 4: Case II: \(2

Here:

\[ f(x)=3-x \]

Since:

\[ 0\le3-x<1 \]

Use first rule:

\[ f(3-x)=1+(3-x)=4-x \]

---

🚀 Exam Shortcut

• Find inner function first
• Check which interval the output belongs to
• Apply piecewise rule again carefully

Spread the love