1. **Problem Statement:** Simplify and understand the concepts of one-to-one functions and inverse functions. 2. **One-to-One Functions:** A function $f$ is one-to-one if for every $x_1$ and $x_2$ in the domain, $f(x_1) = f(x_2)$ implies $x_1 = x_2$. This means no two different inputs map to the same output. 3. **Example:** Is $f(x) = 3x - 1$ one-to-one? - Assume $f(x_1) = f(x_2)$: $$3x_1 - 1 = 3x_2 - 1$$ - Simplify: $$3x_1 = 3x_2$$ - Divide both sides by 3: $$x_1 = x_2$$ - Since $x_1 = x_2$, $f$ is one-to-one. 4. **Inverse Functions:** The inverse function $f^{-1}$ reverses the mapping of $f$. If $f(x) = y$, then $f^{-1}(y) = x$. 5. **Finding Inverse Algebraically:** - Given $y = 3x + 2$, interchange $x$ and $y$: $$x = 3y + 2$$ - Solve for $y$: $$3y = x - 2$$ $$y = \frac{x - 2}{3}$$ - So, $$f^{-1}(x) = \frac{x - 2}{3}$$ 6. **Important Note:** A function must be one-to-one to have an inverse that is also a function. 7. **Horizontal Line Test:** A function is one-to-one if no horizontal line intersects its graph more than once. 8. **Summary:** - One-to-one functions have unique outputs for unique inputs. - Inverse functions reverse the input-output pairs. - To find an inverse, swap $x$ and $y$ and solve for $y$. **Final Answer:** The function $f(x) = 3x - 1$ is one-to-one and its inverse is $$f^{-1}(x) = \frac{x + 1}{3}$$ (note: correcting the inverse for $3x - 1$).