1. **Definition of a Function:** A function is a rule that assigns each input exactly one output. For example, $f(x) = x^2$ means for every $x$, you get one $y$ value. 2. **Types of Functions:** - **One-to-One (Injective):** Each output comes from only one input. If $f(a) = f(b)$, then $a = b$. - **Onto (Surjective):** Every possible output is covered by the function. - **Bijective:** Both one-to-one and onto, meaning a perfect pairing between inputs and outputs. 3. **Limit:** The value a function approaches as the input approaches some point. Written as $\lim_{x \to a} f(x) = L$. 4. **Maxima and Minima:** Points where the function reaches highest or lowest values locally. - **Relative (Local) Maxima/Minima:** Highest or lowest points in a small neighborhood. - **Extreme Values:** The absolute highest or lowest points on the entire function. 5. **Point of Inflection:** Where the curve changes concavity (from curving up to down or vice versa). 6. **Concavity:** Describes if the graph curves upward (concave up) or downward (concave down). 7. **Slope:** The steepness of a line, calculated as rise over run. 8. **Tangent Line:** A line that touches the curve at one point and has the same slope as the curve there. 9. **Secant Line:** A line that intersects the curve at two points. 10. **Derivatives:** Measure how a function changes at any point; the slope of the tangent line. 11. **Differentiation:** The process of finding the derivative. 12. **Integral:** Represents the area under the curve of a function. 13. **Integration:** The process of finding the integral. These concepts help us understand and analyze how functions behave in simple, visual, and practical ways.