1. The problem is to find the inverse of a function $f(x)$, which means finding a function $f^{-1}(x)$ such that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$. 2. The general formula or method to find the inverse function is: - Replace $f(x)$ with $y$. - Swap $x$ and $y$ in the equation. - Solve the new equation for $y$. - The resulting expression for $y$ is $f^{-1}(x)$. 3. Important rules: - The function must be one-to-one (bijective) to have an inverse. - The domain of $f$ becomes the range of $f^{-1}$ and vice versa. 4. Example: Find the inverse of $f(x) = 2x + 3$. - Step 1: Write $y = 2x + 3$. - Step 2: Swap $x$ and $y$: $x = 2y + 3$. - Step 3: Solve for $y$: $$x - 3 = 2y$$ $$y = \frac{x - 3}{2}$$ - Step 4: So, $f^{-1}(x) = \frac{x - 3}{2}$. 5. This means the inverse function reverses the effect of the original function. 6. Always check by composing $f(f^{-1}(x))$ and $f^{-1}(f(x))$ to verify they equal $x$. This is the process to find the inverse of a function.