This tutorial explains how to find the inverse of a function analytically. Several worked examples are presented, each with a detailed step‑by‑step solution. Matched practice exercises with final answers are included at the end of the page.
Find the inverse of the linear function
\[ f(x) = 2x + 3 \]Solution
Step 1: Write the function as an equation.
\[ y = 2x + 3 \]Step 2: Solve for \(x\).
\[ x = \frac{y - 3}{2} \]Step 3: Write the inverse function.
\[ f^{-1}(y) = \frac{y - 3}{2} \]Replacing \(y\) with \(x\):
\[ f^{-1}(x) = \frac{x - 3}{2} \]Check
\[ f(f^{-1}(x)) = 2\left(\frac{x - 3}{2}\right) + 3 = x \] \[ f^{-1}(f(x)) = \frac{(2x + 3) - 3}{2} = x \]Conclusion:
\[ f^{-1}(x) = \frac{x - 3}{2} \]Matched Exercise 1
Find the inverse of the function: \[ f(x) = -x - 4 \]Find the inverse of
\[ f(x) = (x - 3)^2, \quad x \ge 3 \]Solution
\[ y = (x - 3)^2 \]Solving for \(x\) gives two solutions:
\[ x = 3 + \sqrt{y}, \quad x = 3 - \sqrt{y} \]Since \(x \ge 3\), we select the positive branch.
\[ f^{-1}(y) = 3 + \sqrt{y} \]Replacing \(y\) with \(x\):
\[ f^{-1}(x) = 3 + \sqrt{x} \]Check
\[ f(f^{-1}(x)) = (\sqrt{x})^2 = x \] \[ f^{-1}(f(x)) = 3 + \sqrt{(x - 3)^2} = 3 + |x - 3| = x \]Conclusion:
\[ f^{-1}(x) = 3 + \sqrt{x} \]Matched Exercise 2
Find the inverse of the function: \[ f(x) = (x + 1)^2, \quad x \ge -1 \]Find the inverse of
\[ f(x) = \frac{x + 1}{x - 2} \]Solution
\[ y = \frac{x + 1}{x - 2} \]Multiply both sides by \(x - 2\):
\[ y(x - 2) = x + 1 \] \[ yx - 2y = x + 1 \] \[ x(y - 1) = 1 + 2y \] \[ x = \frac{1 + 2y}{y - 1} \]Interchanging \(x\) and \(y\):
\[ f^{-1}(x) = \frac{1 + 2x}{x - 1} \]Matched Exercise 3
Find the inverse of the function: \[ f(x) = \frac{x + 1}{x - 1} \]Find the inverse of
\[ f(x) = \ln(x + 2) - 3 \]Solution
\[ y = \ln(x + 2) - 3 \] \[ \ln(x + 2) = y + 3 \] \[ x + 2 = e^{y + 3} \] \[ x = e^{y + 3} - 2 \] \[ f^{-1}(x) = e^{x + 3} - 2 \]Check
\[ f(f^{-1}(x)) = \ln(e^{x + 3}) - 3 = x \]Conclusion:
\[ f^{-1}(x) = e^{x + 3} - 2 \]Matched Exercise 4
Find the inverse of the function: \[ f(x) = 2\ln(x + 4) - 4 \]
Inverse Function Calculator
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