General Mathematics PDF - Functions & Their Graphs

GENERAL MATHEMATICS -Make sure you look for minimum and maximum values of y. CHAPTER 1:FUNCTIONS &THEIR GRAPHS Example: y= √x+4 Lesson 1: Functions...

GENERAL MATHEMATICS -Make sure you look for minimum and maximum values of y. CHAPTER 1:FUNCTIONS &THEIR GRAPHS Example: y= √x+4 Lesson 1: Functions -We notice the curve is either on or above the horizontal axis. No matter what value of x we try, Function- a relation in which each element of the we will always get a zero or positive value of y. domain corresponds to exactly one element of range; a We say the range in this case is y ≥ 0. relation where each element in the domain is related to * The curve goes on forever vertically, beyond only one value in the range by some rule; is a set of what is shown on the graph, so the range is all non- ordered pairs (x,y) such that no two ordered pairs have negative values of y the same x-value but different y-values. Using Relation- a rule that relates values from a set of functional notation, we can write f(x) = y, read as “f of values (called the domain) to a second set of values x is equal to y.” In particular, if (1, 2) is an ordered pair (called the range); set of ordered pairs (x,y). associated with the function f, then we say that f(2) = 1 Example: Determine whether the following are functions a) A = {(1, 2), (2, 3), (3, 4), (4, 5)}- FUNCTION b) B = {(1, 3), (0, 3), (2, 1), (4, 2)}- FUNCTION Relation in table Relation in graph c) C = {(1, 6), (2, 5), (1, 9), (4, 3)}- not a function because the first element, 1, is repeated. Domain- set of the first coordinates or the x( the set D is the domain of f) When finding the domain, remember: -The denominator (bottom) of a fraction cannot be zero -The number under a square root sign must be positive in this section Example: y= √x+4 Relation in mapping diagram The domain of this function is x≥ −4, since x cannot be less than −4 NOTE! -The only ones that "work" and give us an answer are the ones greater than or equal to −4. This will make the number under the square root positive. -The enclosed (colored-in) circle on the point (−4,0). This indicates that the domain "starts" at this point. *In general, we determine the domain of each function Functions Defined by Equation by looking for those values of the independent variable Defining a function by displaying the rule of (usually x) which we are allowed to use. (Usually we correspondence in a table or listing all the have to avoid 0 on the bottom of a fraction, or negative ordered pairs only works if the domain and values under the square root sign). range are finite. Range- set of the second coordinates or the y ( the set R Vertical Line Test for a Function - if each is the range of f) vertical line in a rectangular coordinate system -The range of a function is the spread of possible passes through at most one point on the graph of the equation but if it passes through two or more y-values (minimum y-value to maximum y-value) points on the graph then the equation does not -Substitute different x-values into the expression define a function for y to see what is happening. (Ask yourself: Is y Domain of a Function- set of all inputs for the always positive? Always negative? Or maybe not function; may be stated explicitly equal to certain values?) Piecewise Function- known as compound function; defined by multiple sub-functions where each sub-function applies to a certain interval of Example: the main functions of the domain Example: Find (g – f)(x). What is h(−1)? x is ≤ 1, so we use h(x) = 2, so h(−1) = 2 Example: What is h(1)? x is ≤ 1, so we use h(x) = 2, so h(1) =2 What is h(4)? x is > 1, so we use h(x) = x, so h(4) =4 Floor Function- a very special piecewise function. It has an infinite number of pieces Example: Example: Lesson 1.2: Evaluating a Function Evaluating a function- finding the value of the function for a given value of the variable Example: f(x)=x^2−5x+3 In terms of function notation we will “ask” this using the notation f(4). So, when there is something other than the variable inside the parenthesis we are really asking what the value of the function is for that particular quantity. Lesson 1.4: Composition of Functions Composition of Functions - another method of * f(4)=(4)2−5(4)+3=16−20+3= −1 contributing a function from two given functions; consists of using the range element of one function Lesson 1.3: Operations on a Function as the domain element of another function Operations on a Function- for all the values of x for The composite of function or composition of f by which both g(x) and h(x) are defined g, is given by (g o f) (x)=g[f(x)] for all x in the SUM: (g + h)(x)= g(x)+h(x) domain of f such that f(x) is in the domain of g DIFFERENCE: (g-h)(x)= g(x)-h(x) PRODUCT: (gh)(x)= g(x). h(x) QUOTIENT: (g/h)(x)= g(x)/h(x); h(x) #0 Example: (g º f)(x) f(x) = 2x+3 and g(x) = x2 *(g º f)(x) = (2x+3)2 (f º g)(x) = f(g(x)) * (f º g)(x) = 2x2+3 Lesson 2.1: Rational Equations Rational Equations- equations that contain rational expressions; it can be solved using the We need regions that make the rational expression techniques for performing operations with rational negative. That means the middle region. Also, expressions and for solving algebraic equations since we’ve got an “or equal to” part in the Example 1: Solve: 5/x−1/3=1/x. inequality we also need to include where the Solution: We first make a note that x≠0 and then inequality is zero, so this means we include x=−1. multiply both sides by the LCD, 3x: Notice that we will also need to avoid x=5 since that gives division by zero. The solution for this inequality is, −1≤x