6 precalculo

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PRECALCULUS OBJECTIVE DATE 06 Understand and analyze functions 09 Jul 2026 and their behavior before studying Functions, Graphs and Their Properties VERSION 1.0 1 FUNCTIONS AND NOTATION 2 DOMAIN AND RANGE 3 FUNCTION TYPES A function assigns each input (x) exactly one output (y). Domain (D): set of all possible input 47 Linear Notation: Range (R): set of all possible output values. independent variable (input) / y: dependent variable (output) Quadratic Examples: Input (x) Output (f(x)) f(x) D 0) R no) f(x) a 0 -2 -3 0 0 Polynomial -1 -1 g(x) f(x) 0 1 h(x) 1 3 Rational 2 5 1 p(x) f(x) f(x) q(x) Example: f(x) 2x Radical ! Domain restrictions come from: denominators 0, even roots 2 logarithms and other (even 4 FUNCTION TRANSFORMATIONS 5 COMPOSITE AND INVERSE FUNCTIONS Let y f(x) be parent function Composite Function f(x) Transformation Equation Effect Graph Do first, then f. Logarithmic Vertical shift up Up units Example: Vertical shift down Down units 6 EXPONENTIAL FUNCTIONS Horizontal shift left Inverse Function Left units Example: f(x) (x) reverses f(x). Horizontal shift right 2 Right units a>1 Replace f(x) with Decreasing Vertical stretch Stretch Domain: 2 Swap and Range: 2 Compress Solve for y. y Horizontal Horizontal 0 Vertical compression 3 Rename as 2 Reflection over x-axis 3 Reflection over y-axis ! A function has an inverse only if is one (passes the horizontal line test). transformed Properties: 7 LOGARITHMIC FUNCTIONS 8 IMPORTANT LIMITS (INTUITIVE) 9 COMMON FUNCTION PROPERTIES f(x) - > a a>1 lim f(x) L Property Definition Test Description approaches as approaches Increasing Decreasing Even Symmetric about (0, . Domain: Range: Range: lim f(x) approaches Odd Symmetric about Vertical 0 Vertical increases without the log, 0 log, One-to-one Passes the (Injective) horizontal line lim f(x) approaches as decreases without bound (1,0) (1,0) If then Graph rises from left / lim f(x) 00 Vertical at If then Graph falls from Decreasing Properties: (xy) left These ideas are the foundation of 10 PARENT FUNCTIONS SUMMARY 11 COMMON MISTAKES Constant Linear Quadratic Cubic Absolute Value Square Root Exponential Logarithmic X Forgetting domain restrictions (e.g., - f(x) Confusing with (a>1) (a>1) X Not checking one to one before finding inverse. ! X Mixing up horizontal vs. vertical shifts. X asymptotes in rational, exponential and logarithmic CONNECTION TO NEXT TOPIC APPLICATIONS - STUDY TIP REMEMBER Precalculus prepares you for limits, . Practice graphing by hand, Every complex model derivatives and integrals by building use different colors, and starts with simple understanding of functions and their Physics Engineering Economics Computer Science explain your reasoning behavior. Growth) Signals) (Optimization) out COLECCION: MATEMATICAS PARA NANOCIENCIAS HOJA 06 DE 13 I VERSION: 2026-07-09