{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

16a_functions - UNIVERSITY OF CALIFORNIA AT DAVIS Dr H Xiao...

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: UNIVERSITY OF CALIFORNIA AT DAVIS Dr. H. Xiao DEPARTMENT OF MATHEMATICS FUNCTIONS DEFINITION: A function is a relationship between two variables such that to each of the values of one variable (called the independent variable) there corresponds exactly one value of the other variable (called the dependent variable). DOMAIN : the set of all values of the independent variable for which the function is defined RANGE: the set of all values taken by the dependent variable EXAMPLES: c = 2m, y = x3+3, F : §C+32 ,f(m) = |m+1|, f(;r) : x/a:2 ~ 1, flat) = $713 Is :9 Is A FUNCTION OF 2:? x2 + y = 1; l>)rc -— y2 = 1 0) THE VERTICAL LINE TEST: Graph the equation with the horizontal axis representing the in— dependent variable. If EVERY vertical line in the plane intersects the equation at most once7 then the equation defines a function. FUNCTION NOTATION: f(r) = 2m, f(x) 2: m3 + 3, f(C) = %C + 32 f(.) z (.)2 - 1 FINDING THE DOMAIN AND RANGE OF THE FOLLOWING FUNCTIONS: f($)=|x+1|, f(m):vx§~1, f($)=-,zlfi (1' DEFINITION: A function is one—t0~0ne if to each value of dependent variable there corresponds exactly one value of the independent variable THE HORIZONTAL LINE TEST: Graph a function with the horizontal axis representing the in- dependent variable. If EVERY horizontal line in the plane intersects the graph at most once, then the function is 0ne~to—one. COMPOSITE FUNCTION: The fungtlbn given by (f o g)(:c) = f (g(a:)) is the composite of f with g. The domain of (fog) is the set of all a: in the domain of 9 such that 9(33) is in the domain Of f. EXAMPLES: f(;t)=\/x—1, g(a:):m2+3, (fog)(x)=\/(x2+3)—1= x2+2 , (gof)=(\/;:T)2+3=$+2 INVERSE FUNCTION : If f and g are two functions such that (f o g)(a¢) = a; for each a: in the domain of g, and (g o f )(15) = a: for each x in the domain of f. Then g is the inverse of f , and is denoted by g :- f’l. EXAMPLES: f(x) : x/x w 1, f“1 = x2 + 1. Verification: (fof‘l) : \/T2 + 1 —1:|:r|: T (since :c Z 1!!), and (f‘lof) = (x/x — 1)2+1 = x iilH.:3'.l‘," ...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern