Determining Symmetry in Functions: Even, Odd, or Neither

RECALL: From your earlier math courses, you have seen functions that are symmetrical (mirror image) about the y-axis such as the function y=x2: Mirror 45 4 35 3 25 1.5 0.5 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 NEW: There are a number of different terms used to describe the ways in which functions behave, Today, you will learn how to determine whether or not a function's symmetrical behavior is even or odd. neither. CASE 1: EVEN FUNCTIONS An EVEN function is reflected about the y-axis. (signs are the same) An even function satisfies f(-x)=f(x) = for all x in its domain Therefore, a function is even if it is unchanged when "x" is replaced by "-x". Example 1: Is f(x) = x2 an even function? Let's Check as signs change . f(x) = x2 f(-x) = (-x)2 f(-x) = x2 f(-x) = f(x)

Therefore, f(x) = x ² is an even function since f(x) = f(-x) y-axis the mirror y 45 4 3.5 3 25 2 1.5 I f(-x) 1 f(x) 0.5 x -3 -2 x -1 0 1 x 2 3 CASE 2: ODD FUNCTIONS An ODD function is: reflected about the origin refected about the line y=x reflected in both the x-axis and y-axis (signs are all different) An odd function satisfies f(-x)=-f(x) = All signs for all x in its domain change - Example 2: Is f(x) = x 3 an odd function? Let's Check f(x) = x ³ f(-x)=(-x)3 = f(-x)=-x3 = f(-x)=-f(x) =

Therefore, f(x) = x ³ is an odd function since f(-x)=-f(x) y y-x f(x) X CASE 3: NEITHER ODD NOR EVEN f(-x)#f(x) At least one f(-x)#-f(x) but not all signs will Example 3: Is f(x) = x2 - X odd, even or neither? (signs vary) f(x)=x2-x change . f(-x)=(-x)2- = - (-x) f(-x)=x2+x Therefore, f(-x) #f(x) > ((x) = x 2 - is not even. odd. Therefore, we can conclude that f(x) = x2 - -xis NEITHER odd nor even. x I