Find \(f^{-1}(x)\) for \(f(x)=4x-3\)

📝 Question

Let:

\[ f:\mathbb{R}\to\mathbb{R}, \quad f(x)=4x-3 \]

Find \(f^{-1}(x)\).

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✅ Solution

🔹 Step 1: Check invertibility

The function is linear with non-zero slope (4).

Hence, it is one-one and onto, so inverse exists.

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🔹 Step 2: Let

\[ y=4x-3 \]

Interchange \(x\) and \(y\):

\[ x=4y-3 \] —

🔹 Step 3: Solve for \(y\)

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

🔹 Step 4: Write inverse

\[ f^{-1}(x)=\frac{x+3}{4} \] —

🎯 Final Answer

\[ \boxed{f^{-1}(x)=\frac{x+3}{4}} \]

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🚀 Exam Shortcut

• For \(ax+b\), inverse = \(\frac{x-b}{a}\)
• Swap \(x,y\) and solve
• Divide by coefficient