MATH117 Study Guide - Periodic Function, Even And Odd Functions, Joule
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19 Nov 2012
School
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=f◦g(t)
For f(x) = 1 −sin x, g(x) = x3,
f◦g(x) = f(x3) = 1 −sin x3
g◦f(x) = g(1 −sin x) = (1 −sin x)3
Domain of Composite Functions
For f◦g(x),
R={y|y=f◦g(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
Rf≡Dg
Typically, the inverse of f(x) is written as
g=f−1(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) = x−1
2, find the inverse
y=x−1
2
2y=x−1
2y+ 1 = x
f−1(y)=2y+ 1
Does f−1(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 f−1(x) = √x
f(x) = x2on (−∞,0] has inverse f−1(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)
2−f(−x)
2
=f(x)
2+f(x)
2+f(−x)
2−f(−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+e−x
2+ex−e−x
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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