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# 1.2(ONE) - Math126 1.2 Basics of Functions and Their Graphs...

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Unformatted text preview: Math126. . 1.2 Basics of Functions and Their Graphs Prerequisites for this chapter You should know about numerical expressions, algebraic expressions, and equations. You also should know how to simplify numeric expressions (order of operations and properties of the real number system), algebraic expressions (order of operations, properties of the real number system, and combining like terms), and how to solve equations (properties of equality, zero product property of the real numbers). In this chapter we will introduce a new objects called functions and will study their properties. You will need to use the properties of numeric'and algebraic expressions and equations when working with functions. What is a function? Function: A function is a particular type of relation between two nonempty sets such that for each element in the ﬁrst set there corresponds exactly one element in the second set. The ﬁrst set is called the domain of the function and the second set is called the range of the function. A relation is a correspondence between two sets of objects. If x and y are two elements in these sets and if a relation exists between it and y, then we say that x corresponds to y or that y depends on x, and we write x —) y. ‘3 r; g 2(- P) Bosnia”: A function from X into Y is a relation that associates with each element of X exactly one element of Y. Four ways to represent functions 0 algebraic: f(x) = x2 + 1 equation: 1. Start with the domain as the set of real numbers. 2. If the equation has a denominator, exclude any numbers that give a zero denominator. 3. If the equation has a radical of even index, exclude any numbers that cause the expression inside the radical to be negative. Note: When the domain is not speciﬁed, in this class we assume that the domain is the set of all real numbers that, as inputs, produce real number outputs. Objectives: 1. Find the domain and range of a relation. {(986, Felicia), (98.3,Gabriella), (99.1, Lakeshia)} :3: %asiti 6&3, 99:13 R: Fem/sawtu, Lamina-j 2. Determine whether a relation is a function. «Lunatiwt ”’25 {(2,3),(2,4),(2,5)} N 0 menace} 995 >62! or 3 RAM} _. ~ ‘8 x+fly—8 531—»(‘443 I: a a x2+2y210 «2— -)(2-+l(i.. x2+y2=l6 L3” _. L E}:’*‘J.rx H4, Not all equations with the variables X and y deﬁne a function. If an equation is solved for y and more than one value of y can be obtained for a given x, then the equation does not deﬁne y as a function of X. So the equation is not a function. 4. Evaluate a function. f is the name of the function (functions need not be named “f”. (X is the input variable independent, variable or argument of the function)f(x) is the output variable, dependent variable or the value of f at x Find a3.) for f(x)=2x2 —4 4 (3'): Z (3)}?— ‘r : l LT Find £02) for f(x)=9-X2 19C“ 2);. g , (—2,) L :: 9—9 :2 6— Find f(x+2), iff(x)=—x2+2x+5. __ J} [x+2)— , -—-(x +2)3‘+ 30“” Q) 1‘ '9 xl-HMH} +§K+L++b z __ - ”(x 2"...th L2+ax+q+b :- -X—‘3\Y+'3 Find f(3(2)) if f(x)= x +3x— 2 and g=(x) 2x 6. i041): L" hat-2r; 9(1): 1(1) :6 e—- :2 : hlr- 66 : --l+ - "Q 33 2 Iff(x)= xx: x,ﬁnd f(—3). 3 a“ (-3): 30'5”““35 U3); -‘B’l-‘é : #23“ r: 23l— —-:L4 #1? C’) 5. Graph functions by plotting points. Graph the following functio 5 f(x) =3x—1 and g(x) =3x 4— 1— T . «1005??ng E .1 6. Use the vertical line test to identiﬂ functions. are iaarat Lha rat as Maria If any vertical time» interstate a graph in mere was sat: paint, the: graph {iﬂﬁﬁ ant ﬁrm: yas a mastic-a air. ___ ‘ 7. Obtain information about a function from its ra h You can obtain information about a function from its graph. At the right or left of a graph you will ﬁnd closed dots, open dots or arrows E 1 0A closed dot indicates that the graph does not extend beyond this point, and the point belongs to the graph. ( ) oAn open dot indicates that the graph does not extend beyond this point and the point does not belong to the graph. éwﬁ ,_ ,,r_ ago-o o up \ +v —~> An arrow indicates that the graph extends indeﬁnitely in the direction in which the arrow points. Analyze the graph of this function f (x) - x2 — 3x— 4 a. Is this a [email protected] - b Find r(4) 4’0» LVBW Lt “L'Ll c. F1ndf(l) 43(1):! rguyir 2. d. For what value of X is f(X)=-4 “1+: igx—Lr ‘Hl “r O ': XL—BX O f: XLYr-B) 8. Identiﬁ: the domain and the range of a function from Itsgraph ' 3: [Eli')3) I_+ l— Raffiigj J : 7 . w r . L —l_l7_k—[ f" T7 ALA T, ,l , f f .; . . _ x . '7 +7 “ -_—l i— 4——| 4—“ " _| L— J _| l—ner L_J_ LT +4 r s 4 L J 9. Identiﬂ the intercepts from a function’s graph. We can identify X and y intercepts from a function's graph. To ﬁnd the X-intercepts, look for the points at which the graph crosses the X axis. The y-intercepts are-the points Where the graph crosses the y axis. The zeros of a function, f, are the X values for which f(X)=0. These are the X intercepts. By deﬁnition of a function, for each value of X we can have at most one value for y. What does this mean in terms of intercepts? A function can have more than one X-intercept but at most one y intercept. ...
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