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Transcript

Chapter 4 Functions and Relations Notes Relation – Any set that can be written as an ordered pair (x, y) Domain - All of the x values of a relation Range – All of the y values of a relation that correspond to the domain Finite Set – Fixed number of ordered pairs Infinite Set – A set that has a never ending set of ordered pairs **** If no set or domain is not specified, the domain would always be the real numbers! FUNCTION - A relation that has one and only one y value for every x value in the domain. Functions - 1. A set is a function if it has no repeating x values 2. A Graph is a function if it passes the vertical line test 3. Even though a graph may not be a function, you can Restrict the domain of the graph to make it a function Function Notation - y = , F(x), F:, F={(x,y)|, Types of function - 1. Linear Functions - y = mx + b, Domain and Range Is always real numbers except x = or y = lines 2. Quadratic Functions - y ax bx c , Domain Is always the Real numbers and the range depends on the turning point. 2 3. Polynomial Functions - y ax ......... Where n is greater than 2. Domain is always The real number and range depends on the power of n. n 4. Absolute Value Function – y = |x| , Domain is All on the real numbers and the range is the Positive real numbers. 5. Square Root Functions - y x , Domain is the Positive real numbers and the range is also the Positive real numbers 6. Rational Functions – y x3 , Domain depends x4 On the denominator and the range follows. 7. Rational Functions with a square root combination of 5 and 6. 8. Identity Function – y=x Direct Variation - The variables are directly proportional to each other. Function Transformations 1. Addition/subtraction of a Constant - Moves function up and down 2. Multiplying a function by a constant – a. Constant greater than 1 - function becomes steeper b. Constant less than 1 – slope of function decreases 3. Absolute Value of a function - The negative part of the function reflects over the x-axis. 4. Add or Subtract a constant in the function - Moves the function right or left. One to one Functions - Has one y for every x and one x for every y. Example: lines, square roots, some polynomial functions Composite Functions - One Function inside of the other function. Symbol - ( f g )( x) f ( g ( x)) Inverse Functions – If you do the composite of two functions and it comes out to x. Simple definition: Switch your x and y and resolve for inverse. Limiting domain – To find the inverse of functions that are not one to one the domain must be limited to create an inverse that is a function. Circles Basic formula – Center at (0,0) x 2 y 2 r 2 , where r = the radius Center – Radius Formula – x h y k 2 2 r 2 , where (h,k) is the Center and r = radius of the circle. Standard Form of Circles – x2 y 2 dx ey f 0 , Basically the Center-Radius form multipled out Changing from the standard form to the Center radius form: Use Completing the Square twice to change the form. Changing from the Center radius form to the standard form: FOIL out the binomials and move all terms to one side. Inverse Variation - Inversely proportional. xy c , where c is a constant The equation is a Hyperbola.