MATH117 Study Guide - Periodic Function, Even And Odd Functions, Joule

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19 Nov 2012
Department
Course
Professor
MATH 117 - Calculus 1 for Engineers
Kevin Carruthers
Fall 2012
Functions
Afunction is a rule which assigns a single output to a variable given at least one input
variable. In this class we deal only with a single input x and write y = f(x)
x is the indepenant variable
y is the dependant variable
Adomain is the set of allowable values for the independant variable (x)
Arange is the set of possible values for the dependant variable (y)
Function Composition
If y=f(x) and x=g(t) then we can view y as a function of t, y=f(g(t)), which we
sometimes write as y=fg(t)
For f(x) = 1 sin x, g(x) = x3,
fg(x) = f(x3) = 1 sin x3
gf(x) = g(1 sin x) = (1 sin x)3
Domain of Composite Functions
For fg(x),
R={y|y=fg(x), x  Dg}
or
R={y|y=f(x), x  Rg}
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Inverse Functions
We say gis the inverse of fif
g(f(x)) = x
for all xin the domain of f. If g(f(x)) = x, then
RfDg
Typically, the inverse of f(x) is written as
g=f1(x)
Note that this does not mean the reciprocal:
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f(x)= (f(x))1
Finding Inverses
In simple cases, we just solve y=f(x) for x.
Example: if f(x) = x1
2, find the inverse
y=x1
2
2y=x1
2y+ 1 = x
f1(y)=2y+ 1
Does f1(x)always exist?
No, f(x) only has an inverse if it is one-to-one, ie. if its graph passes the horizontal line test.
Mathematically, if f(x0) = f(x1), x0=x1
We can restrict the domain so a non-invertible function has an inverse:
f(x) = x2on [0,) has inverse f1(x) = x
f(x) = x2on (−∞,0] has inverse f1(x) = x
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Even and Odd Periodic Functions
A function is even if f(x) = f(x), this means its graph is symmetric about the y-axis.
A function is odd if f(x) = f(x), this means its graph is symmetric about the origin (ie.
reflected in the x- and y-axes).
Most functions are neither even nor odd, but we can transform any function into the sum of
even and odd parts.
f(x) = f(x) + f(x)
2f(x)
2
=f(x)
2+f(x)
2+f(x)
2f(x)
2
=f(x) + f(x)
2+f(x)f(x)
2
=g(x) + h(x)
where g(x) is an even function and h(x) is an odd function.
This idea is used to form two useful functions.
ex=ex+ex
2+exex
2
= cosh x+ sinh x
Usefulness of Symmetry
To sketch f(x) = x2− |x|, we note that f(x) is even, thus we can consider only x > 0 and
use symmetry to determine x < 0
Periodicity
A function f(t) is periodic with period Pif
f(t+nP ) = f(t), n  Z
Any function which exists only over a finite interval can be extended as an even or odd
periodic function.
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