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Find the Inverse of f(x) = 3x − 2

publish date: 2026/05/23 21:16:56.232605 UTC

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Using the three-step process, find \(f^{-1}(x)\) for \(f(x) = 3x - 2\), \(x \in \mathbb{R}\).

\(f^{-1}(x) = 3x + 2\)

\(f^{-1}(x) = \dfrac{x - 2}{3}\)

\(f^{-1}(x) = \dfrac{x + 2}{3}\)

\(f^{-1}(x) = \dfrac{1}{3x - 2}\)

\(f^{-1}(x) = \dfrac{x + 2}{3}\)

Step 1: \(y = 3x - 2 \Rightarrow 3x = y + 2 \Rightarrow x = \frac{y+2}{3}\).

Step 2: \(f^{-1}(y) = \frac{y+2}{3}\).

Step 3: Replace \(y\) with \(x\): \(f^{-1}(x) = \frac{x+2}{3}\).

Verify: \(f(f^{-1}(x)) = 3 \cdot \frac{x+2}{3} - 2 = x + 2 - 2 = x\) ✓

Introduction to Differential Calculus (Systematic Studies with Engineering Applications for Beginners) - 2012