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Graphs and Functions(Review)
MATH 207
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Distance Formula
Example: Find distance between (-1,4) and (-4,-2).
Answer: 6.71
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Midpoint Formula
Example: Find the midpoint from P1(-5,5) to P2(-3,1).
Answer: (-4,3)
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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
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Definition of a Function
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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.
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x
y
Not a function.
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x
y
Function.
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2For the function defined by 3 2 , evaluate:f f x x x
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(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
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2
4(a)
2 3
xf x
x x
2(b) 9g x x
(c) 3 2h x x
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Properties of Functions:
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)
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Determine whether each graph given is an even function, an odd function, or a function that is neither even nor odd.
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32h x x x
35 1g x x
23 24 xxxf
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Where is the function increasing?
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Where is the function decreasing?
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Where is the function constant?
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Local Maxima and Minima
Local Max
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Local Min
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Average rate of change of a Function
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21Find the average rate of change of :
2f x x
From 0 to 1
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Library of Functions (Famous Functions)
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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:
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Application problem:
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Graphing Functions:
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
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2Graph the function 2 3f x x
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.
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
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Theorem
The graph of a function f and the graph of its inverse are symmetric with respect to the line y = x.
f 1
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2 0 2 4 6
2
2
4
6 f
f 1
y = x
(2, 0)
(0, 2)