# 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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