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:
\[ f(x) = 2x + 3 \]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 and swap variables.
\[ f^{-1}(x) = \frac{x - 3}{2} \]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:
\[ f(x) = (x - 3)^2, \quad x \ge 3 \]Write as an equation:
\[ y = (x - 3)^2 \]Solving for \(x\) gives two branches:
\[ x = 3 + \sqrt{y}, \quad x = 3 - \sqrt{y} \]Since \(x \ge 3\), we select the positive branch. 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:
\[ f(x) = \frac{x + 1}{x - 2} \]Write as an equation:
\[ y = \frac{x + 1}{x - 2} \]Multiply both sides by \(x - 2\) and expand:
\[ y(x - 2) = x + 1 \implies yx - 2y = x + 1 \]Gather terms with \(x\) on one side and solve:
\[ x(y - 1) = 2y + 1 \implies x = \frac{2y + 1}{y - 1} \]Interchanging \(x\) and \(y\):
\[ f^{-1}(x) = \frac{2x + 1}{x - 1} \]Matched Exercise 3: Find the inverse of the function \( f(x) = \frac{x + 1}{x - 1} \).
Find the inverse of:
\[ f(x) = \ln(x + 2) - 3 \]Write as an equation and isolate the logarithm:
\[ y = \ln(x + 2) - 3 \implies \ln(x + 2) = y + 3 \]Convert to exponential form and solve for \(x\):
\[ x + 2 = e^{y + 3} \implies x = e^{y + 3} - 2 \]Swap variables:
\[ f^{-1}(x) = e^{x + 3} - 2 \]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 \( f(x) = 2\ln(x + 4) - 4 \).