Inverse of a Rational Function

Step‑by‑step algebra solver for rational functions of the form \( f(x) = \dfrac{ax + b}{cx + d} \). Generate random functions, see detailed algebraic steps with actual numbers, and visualize the function and its inverse reflected across \( y = x \).

📐 Interactive Calculator • Purple Theme • Rational Functions

✨ A rational function \( f(x) = \dfrac{ax + b}{cx + d} \) (with \( ad - bc \neq 0 \) to ensure one-to-one) has an inverse that is also rational. To find the inverse: swap \(x\) and \(y\), then solve for \(y\). Below, generate random examples, see detailed steps, and visualize symmetry.
\( f(x) = \dfrac{2x - 3}{x + 1} \)

📖 Step-by-Step Solution

STEP 1: Replace \(f(x)\) by \(y\)
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STEP 2: Solve for \(x\)
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STEP 3: Interchange \(x\) and \(y\)
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STEP 4: Write the inverse function \(f^{-1}(x)\)
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📊 Graphical meaning
\(f\) (green) and \(f^{-1}\) (blue) are symmetric about \(y = x\) (red).

Adjust axis scale to explore reflection property. Note: Vertical asymptotes may appear.