1. **Function Definition:** A function is a relation where each input has exactly one output. It is often written as $f(x)$, where $x$ is the input and $f(x)$ is the output. 2. **Evaluation of a Function:** To evaluate a function, substitute the input value into the function's formula and simplify. For example, if $f(x) = 2x + 3$, then $f(2) = 2(2) + 3 = 7$. 3. **Domain / Range of a Function:** The domain is the set of all possible input values ($x$) for which the function is defined. The range is the set of all possible output values ($f(x)$). 4. **One-to-One, Onto, and One-to-One Correspondence:** - One-to-one (injective): Each output corresponds to exactly one input. - Onto (surjective): Every possible output is mapped by some input. - One-to-one correspondence (bijective): Both one-to-one and onto. 5. **Algebraic Combinations of Functions:** Functions can be added, subtracted, multiplied, or divided to create new functions. For example, $(f+g)(x) = f(x) + g(x)$. 6. **Function Compositions:** The composition of two functions $f$ and $g$ is $(f \circ g)(x) = f(g(x))$. 7. **Inverse of a Function:** The inverse function $f^{-1}$ reverses the effect of $f$, so $f(f^{-1}(x)) = x$. 8. **Even / Odd Functions:** - Even functions satisfy $f(-x) = f(x)$ and are symmetric about the y-axis. - Odd functions satisfy $f(-x) = -f(x)$ and are symmetric about the origin. 9. **Six Parent Graphs:** These include basic graphs of linear, quadratic, cubic, absolute value, square root, and reciprocal functions. 10. **Transformation:** Functions can be shifted, stretched, compressed, or reflected to create new graphs. 11. **Quadratic Functions:** Functions of the form $f(x) = ax^2 + bx + c$ with parabolic graphs. 12. **Polynomial Functions:** Functions that are sums of powers of $x$ with coefficients. 13. **Rational Functions & Asymptotes:** Functions expressed as ratios of polynomials, often with vertical and horizontal asymptotes. 14. **Exponential Functions & Their Graphs:** Functions of the form $f(x) = a^x$ where $a > 0$ and $a \neq 1$. 15. **Logarithmic Functions & Their Graphs:** The inverse of exponential functions, $f(x) = \log_a x$. 16. **Solving Exponential & Logarithmic Functions:** Techniques to find $x$ when it appears in exponents or logarithms. 17. **Trigonometric Functions:** Functions like sine, cosine, and tangent that relate angles to ratios in right triangles. 18. **Introduction to Complex Numbers:** Numbers of the form $a + bi$ where $i^2 = -1$.