( ) is symmetric about the y - axis.

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1 (Section 1.5: Analyzing Graphs of Functions) 1.51 PART F: FUNCTIONS THAT ARE EVEN / ODD / NEITHER; SYMMETRY A function f is even f ( x) = f ( x) x Dom( f ) The graph of y = f x for every x in the domain of f is symmetric about the y - axis. (We will discuss this in the following.) If f ( x) = x 2, then f is even, because x R, f ( x) = ( x) 2 = x 2 = f x y The bowl graph of f x = x2 below is symmetric about the y-axis. This means that the parts of the graph to the right and to the left of the y-axis are mirror images (or reflections) of each other. See Section 1.7, Notes More formally, the point x, y does. point x, y lies on the graph if and only if the

2 (Section 1.5: Analyzing Graphs of Functions) 1.52 The term even function may have come from the following fact: If f ( x) = x n, where n is an even integer, then f is an even function. These are the functions for:, x 4, x 2, x 0, x 2, x 4,. We will discuss these further in Section 1.6. The graph for the x 2 function on the previous page is called a parabola. However, the graphs for x 4, x 6, etc. are not parabolas. We will discuss parabolas more in Chapter 2. The reciprocal of a nonzero even function is even. The functions for both x 2 and x 2 which equals 1 x 2 are even. (also see Chapter 4) The graph of the even function f x = cos x is below. Its reciprocal function, for sec x, is also even.

3 (Section 1.5: Analyzing Graphs of Functions) 1.53 A function f is odd f ( x) = f ( x) x Dom( f ) The graph of y = f ( x) is symmetric about the origin. (We will discuss this in the following.) If f ( x) = x 3, then f is odd, because x R, f ( x) = ( x) 3 = x 3 = f x y The snake graph of f x = x3 below is symmetric about the origin. This means that, if the part of the graph to the right of the y-axis (in black) is rotated 180 about the origin, then the result coincides with the part of the original graph to the left of the y-axis (in blue). In fact, if the entire graph is rotated this way, we obtain it again. If the graph has a y-intercept, it must be at the origin. lies on the graph if and only if the does. In the graph below, imagine rotating the black More formally, the point x, y point x, y dashed line segment 180 about the origin until it coincides with the blue dashed line segment.

4 (Section 1.5: Analyzing Graphs of Functions) 1.54 The term odd function may have come from the following fact: If f ( x) = x n, where n is an odd integer, then f is an odd function. These are the functions for:, x 3, x 1, x 1, x 3, x 5,. We will discuss these further in Section 1.6. The reciprocal of a nonzero odd function is odd. The functions for both x 1 (which equals x) and x 1 which equals 1 x are odd. (also see Chapter 4) The graph of the odd function f ( x) = sin x is below. Its reciprocal function, for csc x, is also odd. Note: tan x and cot x are also odd. Note: The zero function (over various domains that are symmetric about 0) is the only function that is both even and odd. (Can you show this?) Note: Many functions are neither even nor odd.

5 PART G: COMBINING EVEN (OR ODD) FUNCTIONS (Section 1.5: Analyzing Graphs of Functions) 1.55 (We will formally discuss sums and differences of functions in Section 1.8.) A constant multiple of f ( x) has the form cf ( x), where c is some real number. For example, 3x 4 is a constant multiple of x 4. Sums, differences, and constant multiples of even functions are even. Think: E ± E = E, and ce = E. In other words, linear combinations of even functions are even. This idea appears in Calculus and in Linear Algebra. See the on the next page. Sums, differences, and constant multiples of odd functions are odd. Special Cases: Polynomial Functions Let f ( x) be a polynomial in standard form, possibly with terms reordered. If every term of f ( x) has even degree, then it is even. Remember: Nonzero constant terms have even degree (namely, degree 0), so they are permitted in an even f x. If every term of f x Warning: If f x has odd degree, then it is odd. has a nonzero constant term, then it cannot be odd.

6 (Section 1.5: Analyzing Graphs of Functions) 1.56 Show that f ( x) = 3x 4 7x is even. f ( x) = 3( x) 4 7( x) = 3x 4 7x = f x (This is true x R.) Show that f ( x) = 4x 3 + x is odd. f ( x) = 4( x) 3 + ( x) = 4x 3 x = 4x 3 + x = f x (This is true x R.) Is f ( x) = x 3 + x + 1 even, odd, or neither? f ( x) = ( x) 3 + ( x) + 1 = x 3 x + 1 This is equivalent to neither f x is neither even nor odd. f x The odd-degree terms prevent f x constant term +1 prevents it from being odd. nor f ( x) over R, the domain of f, so from being even, and the nonzero

7 (Section 1.5: Analyzing Graphs of Functions) 1.57 Assume that we have an even function and an odd function that are not constantly 0 in value on their domains. That is, assume the functions take on nonzero values somewhere. The sum or difference of the even function and the odd function must be neither even nor odd. (For a challenge, experiment with graphs to see why this is true.) Think: nonzero E + ( nonzero O) = neither. In Calculus: The ability to identify even and odd functions comes in very handy in Calculus, especially when it comes to graphing (because of the symmetry properties) and definite integration. Procedures for finding areas and volumes that involve definite integration can be made more efficient. You can also save yourself a great deal of work sometimes in Calculus III: Math 252 at Mesa for example, in finding centers of mass. Technical Note: The even-or-odd rules for products and quotients follow the rules for multiplying and dividing signed real numbers, provided you let + represent even and represent odd. For example, (even)(odd) = (odd), because (+)( ) = ( ). When in doubt, do as we have done in the s: Find and simplify f ( x), and compare it to f x and f ( x).