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MATH 141. Chapter 1: Graphs and Functions (Review). Distance Formula. 1.1. Example: Find distance between (-1,4) and (-4,-2). Answer: 6.71. Midpoint Formula. Example: Find the midpoint from P 1(-5,5) to P 2(-3,1). Answer: (-4,3). y. ( x , y ). r. ( h , k ). x. 1.2. - PowerPoint PPT Presentation
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The standard form of an equation of a circle with radius r and center (h, k) is:
The Unit Circle equation is:
x
y
(h, k)
r(x, y)
Equations in two variables – Example: Circle
Equations
222 rkyhx
222 ryx
1.2
Theorem: Vertical Line TestA set of points in the xy - plane is the graph of a function if and only if a vertical line intersects the graph in at most one point.
(a) For each x in the domain of f, there is exactly one image f(x) in the range; however, an element in the range can result from more than one x in the domain.
(b) f is the symbol that we use to denote the function. It is symbolic of the equation that we use to get from an x in the domain to f(x) in the range.
(c) If y = f(x), then x is called the independent variable or argument of f, and y is called the dependent variable or the value of f at x.
SummaryImportant Facts About Functions
Properties of Functions:1.4
Even and Odd Functions
A function f is even if for every number x in its domain the number -x is also in its domain and
f(-x) = f(x)
A function f is odd if for every number x in its domain the number -x is also in its domain and
f(-x) = - f(x)
Determine whether each graph given is an even function, an odd function, or a function that is neither even nor odd.
2
The function is defined as
if < 0
2 if = 0
2 if > 0
(a) Find (-2), (0), and (3). (b) Determine the domain of .
(c) Graph .
f
x x
f x x
x x
f f f f
f
(d) Use the graph to find the range of .
(e) Is continuous on its domain?
f
f
Piecewise-defined Functions:Example:
Graphing Functions:1.6
2
2
2
Use the graph of to obtain the graph of the following:
(a) 2
(b) 2
f x x
g x x
h x x
.
The inverse of f, denoted by f -1 , is a function such that f -1(f( x )) = x for every x in the domain of f and f(f
-1(x))=x for every x in the domain of f -1:
Inverse Functions
1.7
Theorem
The graph of a function f and the graph of its inverse are symmetric with respect to the line y = x.
f 1