Show \(f(x)=4x+3\) is Invertible on \(\mathbb{R}\) and Find \(f^{-1}\)
📺 Video Explanation
📝 Question
Show that:
\[ f:\mathbb{R}\to\mathbb{R},\qquad f(x)=4x+3 \]
is invertible. Find:
\[ f^{-1}(x) \]
✅ Solution
🔹 Step 1: Show that \(f\) is one-one
Assume:
\[ f(x_1)=f(x_2) \]
Then:
\[ 4x_1+3=4x_2+3 \]
Subtract 3:
\[ 4x_1=4x_2 \]
Divide by 4:
\[ x_1=x_2 \]
Therefore:
\[ f \text{ is one-one} \]
🔹 Step 2: Show that \(f\) is onto
Let:
\[ y\in\mathbb{R} \]
We must show there exists:
\[ x\in\mathbb{R} \]
such that:
\[ f(x)=y \]
So:
\[ 4x+3=y \]
Solve:
\[ 4x=y-3 \]
\[ x=\frac{y-3}{4} \]
Since \(y\in\mathbb{R}\), clearly:
\[ x\in\mathbb{R} \]
Hence:
\[ f \text{ is onto} \]
🔹 Step 3: Find inverse function
Let:
\[ y=4x+3 \]
Solve for \(x\):
\[ x=\frac{y-3}{4} \]
Replace \(y\) by \(x\):
\[ \boxed{f^{-1}(x)=\frac{x-3}{4}} \]
🎯 Final Answer
Since \(f\) is bijective, it is invertible.
Therefore:
\[ \boxed{f^{-1}(x)=\frac{x-3}{4}} \]
🚀 Exam Shortcut
- Linear function with non-zero coefficient is invertible
- Solve \(y=f(x)\) for \(x\)
- Swap variables to get inverse