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 Elementary Functions below. Part 1, Functions Both graphs allow us to view the y-axis as a mirror. Lecture 1.4a, Symmetries of Functions: Even and Odd Functions A reflection across the y-axis leaves the function unchanged. This reflection is an example of a symmetry.

Dr. Ken W. Smith

Sam Houston State University

2013

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A symmetry of a function can be represented by an algebra statement. What other symmetries might functions have? Reflection across the y-axis interchanges positive x-values with negative We can reflect a graph about the x-axis by replacing f(x) by −f(x). x-values, swapping x and −x. But could a graph be fixed by this reflection? Whenever a number is equal Therefore f(−x) = f(x). to its negative, then the number is zero. The statement, “For all x ∈ R,f (−x) = f(x)” (x = −x =⇒ 2x = 0 =⇒ x = 0.) is equivalent to the statement So if f(x) = −f(x) then f(x) = 0. “The graph of the function is unchanged by reflection across the y-axis.” 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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So far, we have discussed two types of symmetry for graphs of functions: Definition. A function f(x) is even if f(−x) = f(x). 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). The function is odd if f(−x) = −f(x). We note that functions like f(x) = x2 and f(x) = x4, where the exponent An even function has reflection symmetry about the y-axis. on x is even will have the property that f(−x) = f(x) since −1 to an An odd function has rotational symmetry about the origin. even integer power is equal to 1. We can decide algebraically if a function is even, odd or neither by Similarly, functions like f(x) = x, f(x) = x3 and f(x) = x5, where the replacing x by −x and computing f(−x). 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. If f(−x) = f(x), the function is even. If f(−x) = −f(x), the function is odd.

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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

Even Odd

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Odd Odd Even Odd

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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.

Odd Even

We can check this algebraically: Smith (SHSU) Elementary Functions 2013 11 / 25 Smithf(− (SHSU)x) = (−x)3 − 4(−Elementaryx) = − Functionsx3 + 4x = −(x3 − 4x) = −2013f(x). 12 / 25 Even and odd functions, example 2 Even and odd functions, example 3

x 2 Decide algebraically if the function f(x) = is even, odd, or 5 2 1 + x2 2 Decide algebraically if the function f(x) = x + 7x − 3x + 5 is even, neither. odd, or neither.

Solution. Solution. x −x If f(x) = then f(−x) = . If f(x) = x5 + 7x2 − 3x + 5 then 1 + x2 1 + (−x)2 2 2 Since (−x) = x we can simplify this to f(−x) = (−x)5 + 7(−x)2 − 3(−x) + 5 = −x5 + 7x2 + 3x + 5. −x x f(−x) = = − = −f(x). 1 + (−x)2 1 + x2 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. So f(x) is odd.

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Testing the concepts. Elementary Functions There is a function which is both even and odd! What is it? Part 1, Functions Lecture 1.4b, Symmetries of Functions: Periodic Functions ?? Dr. Ken W. Smith

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For example, if we look at the graphs below, we see graphs that appear to In this lesson we discuss periodic functions and also introduce the greatest represent periodic functions. 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. The graph on the left has period 2π, slightly more than 6. The graph on the right has period 2.

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We digress from our discussion of periodic functions to introduce a function 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 greatest integer 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 with everything to the right of the decimal place stripped away. The graph on the left has period 2π, slightly more than 6. So it is easy to compute bxc when x ≥ 0. The graph on the right has period π, slightly more than 3. One has to be careful if x is negative – we always round down here, so We will look more closely at periodic functions several times in this course. b−1.1c = −2 Smith (SHSU) Elementary Functions 2013 19 / 25 Smith (SHSU) Elementary Functions 2013 20 / 25 Visualizing functions Visualizing functions

Here is a graph of the greatest-integer function. 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) = bxc then f(x) = f(x − 1) + 1.

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

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 − bxc keeps just the remainder, after we remove the integer part.

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