Notes 1.4 Function Graphs

1.4 1.4.1

Symmetries of functions Even and odd functions

A symmetry of a function is a transformation of the function that leaves the graph unchanged. For example, consider the functions f (x) = x2 and g(x) = |x|. Their graphs are drawn in figure 21. Both of these functions have the property that their 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.

Figure 21. Symmetry about the x-axis. In most cases, a symmetry of a function can be represented by an algebra statement. Here, 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 x-axis.” What other symmetries might functions have? We can reflect a graph about the x-axis by replacing f (x) by −f (x). However, it is unlikely that a graph is fixed by this reflection since 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. However, it is possible for us to 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). 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 on 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. 34

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. 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.

(a)

(b)

(c)

(d)

(e)

(f)

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(g)

(h)

(i)

(j)

Solutions. Graphs (a), (c), (g) and (j) are even functions since a reflection across the y-axis preserves symmetry. All the other graphs represent odd functions since a rotation of 180◦ about the origin preserves the graph. 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. Here is the graph:

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We can check this algebraically: f (−x) = (−x)3 − 4(−x) = −x3 + 4 = −(x3 − 4) = −f (x). 2. Decide algebraically if the function f (x) =

x is even, odd, or neither. 1 + x2

Solution. If f (x) =

x −x then f (−x) = . Since (−x)2 = x2 we can simplify this to 2 1+x 1 + (−x)2 f (−x) =

−x x =− = −f (x). 1 + (−x)2 1 + x2

So f (x) is odd. 3. 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. Testing the concepts. There is a function which is both even and odd! What is it? 1.4.2

Periodic functions

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 right 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. For example, if we look at graphs (a), (c), (d) and (e) in the previous collection of graphs, we see graphs that appear to represent periodic functions.

(a)

(c)

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(d)

(e)

The functions with graphs (a) and (d) have period 2π, slightly more than 6. Graph (c) represents a function with period 2 and graph (e) represents a function with period π, slightly larger than 3. We will look more closely at periodic functions several times in this course. 1.4.3

The greatest integer function and other interesting examples

We digress 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. So it is easy to compute bxc when x ≥ 0. One has to be careful if x is negative – we always round down here, so b−1.1c = −2 Here is a graph of the greatest-integer function.

Figure 22. 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 38

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

Figure 23. A graph of the “fractional part” function. 1.4.4

Resources for function symmetries

Function symmetries are often covered in a section or chapter on function transformations. Here are some textbook resources on function symmetries. In the free textbook, Precalculus, by Stitz and Zeager (version 3, July 2011, available at stitz-zeager.com) even and odd functions are covered in section 1.6, the section on function tranformations. (See page 95.) Periodic functions are covered in section 10.5, in the study of trigonometric functions. In the free textbook, Precalculus, An Investigation of Functions, by Lippman and Rassmussen (Edition 1.3, available at www.opentextbookstore.com) even and odd functions are covered in section 1.5, the section on function transformations (see page 71.) Periodic functions are covered in chapter 6 (beginning at page 353) as that textbook moves from triangle trigonometry to circular functions. In the textbook by Ratti & McWaters, Precalculus, A Unit Circle Approach, 2nd ed., c. 2014, here at Amazon.com this material appears in section 1.5, along with the material on transformations. In the textbook by Stewart, Precalculus, Mathematics for Calculus, 6th ed., c. 2012, (here at Amazon.com) this material appears in in section 2.5, along with the material on transformations. (In July 2013 the first textbook was $147 at Amazon.com and the second textbook was $136 at Amazon.com They are even more expensive in campus bookstores.) There are lots of online resources for studying function symmetry. Here are some I recommend. 1. Wikipedia on function parity 2. Paul’s online notes on function symmetry 39

3. Applet and exercises at Khan Academy

Worksheet to go with these notes. As class homework, please complete Worksheet 1.4, Symmetries of functions, available through the class webpage.

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