Find \(f^{-1}\) for \(f(x)=3x\) on Given Finite Sets

📺 Video Explanation

📝 Question

Let:

\[ A=\{0,-1,-3,2\},\qquad B=\{-9,-3,0,6\} \]

and:

\[ f:A\to B,\qquad f(x)=3x \]

Find:

\[ f^{-1} \]

---

✅ Solution

🔹 Step 1: Find image of each element of \(A\)

Using:

\[ f(x)=3x \]

We get:

• \(f(0)=0\)
• \(f(-1)=-3\)
• \(f(-3)=-9\)
• \(f(2)=6\)

So:

\[ f=\{(0,0),(-1,-3),(-3,-9),(2,6)\} \]

---

🔹 Step 2: Check whether inverse exists

All outputs are distinct, so \(f\) is one-one.

Range:

\[ \{-9,-3,0,6\}=B \]

So \(f\) is onto.

Hence:

\[ f \text{ is bijective} \]

Therefore inverse exists.

---

🔹 Step 3: Write inverse function

To find inverse, interchange each ordered pair:

\[ f^{-1}=\{(0,0),(-3,-1),(-9,-3),(6,2)\} \]

---

🎯 Final Answer

\[ \boxed{f^{-1}=\{(0,0),(-3,-1),(-9,-3),(6,2)\}} \]

Therefore, inverse exists.

---

🚀 Exam Shortcut

• Find all output values first
• Check one-one and onto
• Inverse = reverse ordered pairs