All of Pre Calculus Explained in 15 Minutes

Pre-calculus is the essential toolkit for calculus, focusing on understanding function behavior, manipulating expressions, and interpreting graphs. A function is defined as a machine that takes an input (x) and produces exactly one output (y), represented as f(x) = something. Key concepts include domain (possible x-values) and range (possible y-values). Different types of functions, such as linear, quadratic, polynomial, rational, exponential, logarithmic, and trigonometric, each have unique properties.

Functions can be transformed by shifting, stretching, flipping, and compressing. Adding/subtracting 'k' outside f(x) shifts vertically, while adding/subtracting 'h' inside f(x) shifts horizontally (in reverse direction). Multiplying f(x) by 'a' stretches/compresses vertically, and multiplying x by 'b' inside f(x) compresses/stretches horizontally. Negating f(x) flips over the x-axis, and negating x inside f(x) flips over the y-axis. Polynomials are functions with terms of different powers of x, where the highest power is the degree and the leading coefficient determines the end behavior. Zeros (x-intercepts) are found using methods like the quadratic formula for degree 2 polynomials.

Rational functions are ratios of polynomials. They are characterized by asymptotes: vertical asymptotes occur where the denominator is zero, and horizontal asymptotes depend on the degrees of the numerator and denominator. Exponential functions (f(x) = a * b^x) model growth (b > 1) or decay (0 < b < 1), with 'e' being a crucial base in calculus. Logarithms are the inverse of exponentials, written as log_b(n) = y means b^y = n. Important logarithm rules (for multiplication, division, and exponents) simplify solving exponential equations.

Trigonometry on the unit circle defines x as cos(theta) and y as sin(theta), with angles measured in radians. Key angles and their sine/cosine values should be memorized. Essential identities include sin^2(theta) + cos^2(theta) = 1. Reciprocal functions like secant, cosecant, and cotangent are also important. Inverse trigonometric functions (arcsin, arccos, arctan) return the angle for a given sine, cosine, or tangent value, useful for solving trigonometric equations.

Sequences are ordered lists of numbers, with arithmetic sequences having a common difference and geometric sequences having a common ratio. Series are the sums of sequence terms, with specific formulas for arithmetic and geometric series (both finite and infinite). Limits are a foundational concept for calculus, asking what value f(x) approaches as x gets close to a certain number or approaches infinity. Limits are particularly important where direct substitution leads to undefined expressions (e.g., 0/0). Continuity ensures no jumps, holes, or breaks in a function.