ain this lesson we look at even and odd functions ... · function common in mathematics and...

Elementary FunctionsPart 1, Functions

Lecture 1.4a, Symmetries of Functions: Even and Odd Functions

Dr. Ken W. Smith

Sam Houston State University

2013

Smith (SHSU) Elementary Functions 2013 1 / 25

Even and odd functions

In this lesson we look at even and odd functions.A symmetry of a function is a transformation that leaves the graphunchanged.Consider the functions f(x) = x2 and g(x) = |x| whose graphs are drawnbelow.Both graphs allow us to view the y-axis as a mirror.A reflection across the y-axis leaves the function unchanged.This reflection is an example of a symmetry.


Reflection across the y-axis

A symmetry of a function can be represented by an algebra statement.Reflection across the y-axis interchanges positive x-values with negativex-values, swapping x and −x.Therefore f(−x) = f(x).The statement,

“For all x ∈ R, f(−x) = f(x)”

is equivalent to the statement

“The graph of the function is unchanged by reflection across the y-axis.”


Rotation about the origin

What other symmetries might functions have?We can reflect a graph about the x-axis by replacing f(x) by −f(x).But could a graph be fixed by this reflection? Whenever a number is equalto its negative, then the number is zero.

(x = −x =⇒ 2x = 0 =⇒ x = 0.)

So if f(x) = −f(x) then f(x) = 0.But we could reflect a graph across first one axis and then the other.Reflecting a graph across the y-axis and then across the x-axis isequivalent to rotating the graph 180◦ around the origin.When this happens, f(x) = −f(−x).If f(x) = −f(−x) then we have rotational symmetry about the origin.In this case, we may multiply both sides of the equation by −1 and write

f(−x) = −f(x).



So far, we have discussed two types of symmetry for graphs of functions:

1 Reflection symmetry about the y-axis, in which case f(−x) = f(x).

2 Rotation symmetry about the origin, in which case f(−x) = −f(x).We note that functions like f(x) = x2 and f(x) = x4, where the exponenton x is even will have the property that f(−x) = f(x) since −1 to aneven integer power is equal to 1.

Similarly, functions like f(x) = x, f(x) = x3 and f(x) = x5, where theexponent on x is odd will have the property that f(−x) = −f(x) since −1to an odd power is equal to −1. This motivates the following definitions.



Definition.A function f(x) is even if f(−x) = f(x).

The function is odd if f(−x) = −f(x).

An even function has reflection symmetry about the y-axis.An odd function has rotational symmetry about the origin.

We can decide algebraically if a function is even, odd or neither byreplacing x by −x and computing f(−x).

If f(−x) = f(x), the function is even.If f(−x) = −f(x), the function is odd.



Examples. The graphs of a variety of functions are given below (on thispage and the next). Consider the symmetries of the graph y = f(x) anddecide, from the graph drawings, if f(x) is odd, even or neither.

Even Odd



Even Odd



Odd Odd



Even Odd



Odd Even


Even and odd functions, some examples

Three worked exercises.

1 Graph the function f(x) = x3 − 4x and then decide if the function iseven, odd, or neither. Solution. This function is odd since it issymmetric about the origin.

We can check this algebraically:

f(−x) = (−x)3 − 4(−x) = −x3 + 4x = −(x3 − 4x) = −f(x).Smith (SHSU) Elementary Functions 2013 12 / 25

Even and odd functions, example 2

2 Decide algebraically if the function f(x) =x

1 + x2is even, odd, or

neither.

Solution.

If f(x) =x

1 + x2then f(−x) = −x

1 + (−x)2.

Since (−x)2 = x2 we can simplify this to

f(−x) = −x1 + (−x)2

= − x

1 + x2= −f(x).

So f(x) is odd.


Even and odd functions, example 3

2 Decide algebraically if the function f(x) = x5 + 7x2 − 3x+ 5 is even,odd, or neither.

Solution.If f(x) = x5 + 7x2 − 3x+ 5 then

f(−x) = (−x)5 + 7(−x)2 − 3(−x) + 5 = −x5 + 7x2 + 3x+ 5.

Since f(−x) = −x5 + 7x2 + 3x+ 5 is neither equal to f(x) nor equal to−f(x) then f(x) is neither even nor odd.


Even and odd functions: can a function be both??

Testing the concepts.There is a function which is both even and odd! What is it?

??

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Elementary FunctionsPart 1, Functions

Lecture 1.4b, Symmetries of Functions: Periodic Functions

Dr. Ken W. Smith

Sam Houston State University

2013


Periodic functions

In this lesson we discuss periodic functions and also introduce the greatestinteger function.

Some graphs have translation symmetry, that is, we may shift the graphalong the x-axis a certain amount and leave the graph unchanged. In thiscase the function is periodic; there is a real number c so that if we shiftthe graph to the left by c units, then the graph is unchanged.

Algebraically, we write f(x+ c) = f(x).

The smallest positive real number c such that f(x+ c) = f(x) is calledthe period of the function f .

We will see this phenomenon (periodic functions and translationsymmetry) throughout our study of trigonometry.


Visualizing functions

For example, if we look at the graphs below, we see graphs that appear torepresent periodic functions.

The graph on the left has period 2π, slightly more than 6.The graph on the right has period 2.



The graph on the left has period 2π, slightly more than 6.The graph on the right has period π, slightly more than 3.We will look more closely at periodic functions several times in this course.



We digress from our discussion of periodic functions to introduce afunction common in mathematics and computer science.The greatest integer function

f(x) = bxc,takes as input a real number and rounds the number down to the greatestinteger less than or equal to it.

For example, it rounds 3.1 to 3, so b3.1c = 3.

If the input is already an integer, the output is unchanged. For example,b5c = 5.

If the number x is positive, bxc is essentially the value of x witheverything to the right of the decimal place stripped away.

So it is easy to compute bxc when x ≥ 0.One has to be careful if x is negative – we always round down here, sob−1.1c = −2



Here is a graph of the greatest-integer function.



The greatest-integer function is also called the floor function since isrounds down to the integer “on the floor”, below x.

Notice a certain symmetry of this function: if we translate the graph upand to the right (at an angle of 45◦) then we get the same graph back.

In other words, if f(x) = bxc then f(x) = f(x− 1) + 1.



A function related to the greatest-integer function is the fractional-partfunction.

The floor function throws away the decimal part of a positive real number.

What if, instead, we keep only the decimal part?

The fractional-part function g(x) = x− bxc keeps just the remainder,after we remove the integer part.



The fractional-part function is an example of a sawtooth function – it isperiodic with very sharp edges!



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ain this lesson we look at even and odd functions ... · function common in mathematics and...

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