Even and odd functions In this lesson we look at even and odd - - PowerPoint PPT Presentation

Elementary Functions

Part 1, Functions Lecture 1.4a, Symmetries of Functions: Even and Odd Functions

• Dr. Ken W. Smith

Sam Houston State University

2013

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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 graph unchanged. Consider the functions f(x) = x2 and g(x) = |x| whose graphs are drawn below. 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.

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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 negative x-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.”

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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 equal to 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 is equivalent 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).

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Even and odd functions

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 exponent

• n x is even will have the property that f(−x) = f(x) since −1 to an

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

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Even and odd functions

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 by replacing 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.

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Even and odd functions

• Examples. The graphs of a variety of functions are given below (on this

page and the next). Consider the symmetries of the graph y = f(x) and decide, from the graph drawings, if f(x) is odd, even or neither. Even Odd

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Even and odd functions

Even Odd

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Even and odd functions

Odd Odd

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Even and odd functions

Even Odd

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Even and odd functions

Odd Even

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Even and odd functions, some examples

Three worked exercises.

1 Graph the function f(x) = x3 − 4x and then decide if the function is

even, odd, or neither. Solution. This function is odd since it is symmetric about the origin. We can check this algebraically: f(−x) = (−x)3 − 4(−x) = −x3 + 4x = −(x3 − 4x) = −f(x).

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Even and odd functions, example 2

2 Decide algebraically if the function f(x) =

x 1 + x2 is even, odd, or neither. Solution. If f(x) = x 1 + x2 then f(−x) = −x 1 + (−x)2 . Since (−x)2 = x2 we can simplify this to f(−x) = −x 1 + (−x)2 = − x 1 + x2 = −f(x). So f(x) is odd.

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Even and odd functions, example 3

2 Decide algebraically if the function f(x) = x5 + 7x2 − 3x + 5 is even,

• dd, 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.

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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 Functions

Part 1, Functions Lecture 1.4b, Symmetries of Functions: Periodic Functions

• Dr. Ken W. Smith

Sam Houston State University

2013

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Periodic functions

In this lesson we discuss periodic functions and also introduce the greatest integer function. Some graphs have translation symmetry, that is, we may shift the graph along the x-axis a certain amount and leave the graph unchanged. In this case the function is periodic; there is a real number c so that if we shift the 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 called the period of the function f. We will see this phenomenon (periodic functions and translation symmetry) throughout our study of trigonometry.

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Visualizing functions

For example, if we look at the graphs below, we see graphs that appear to represent periodic functions. The graph on the left has period 2π, slightly more than 6. The graph on the right has period 2.

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Visualizing functions

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.

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Visualizing functions

We digress from our discussion of periodic functions to introduce a function common in mathematics and computer science. The greatest integer function f(x) = ⌊x⌋, takes as input a real number and rounds the number down to the greatest integer less than or equal to it. For example, it rounds 3.1 to 3, so ⌊3.1⌋ = 3. If the input is already an integer, the output is unchanged. For example, ⌊5⌋ = 5. If the number x is positive, ⌊x⌋ is essentially the value of x with everything to the right of the decimal place stripped away. So it is easy to compute ⌊x⌋ when x ≥ 0. One has to be careful if x is negative – we always round down here, so ⌊−1.1⌋ = −2

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Visualizing functions

Here is a graph of the greatest-integer function.

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Visualizing functions

The greatest-integer function is also called the floor function since is rounds down to the integer “on the floor”, below x. Notice a certain symmetry of this function: if we translate the graph up and to the right (at an angle of 45◦) then we get the same graph back. In other words, if f(x) = ⌊x⌋ then f(x) = f(x − 1) + 1.

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Visualizing functions

A function related to the greatest-integer function is the fractional-part function. 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 − ⌊x⌋ keeps just the remainder, after we remove the integer part.

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Visualizing functions

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

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Visualizing functions

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