Discuss the concept of even and odd functions graphically and analytically.
## Even FunctionsThe graph of even functions are symmetric with respect to the y axis.To proove analytically that a given function f is even, we need to proove that: f(x) = f(-x)## Odd FunctionsThe graph of even functions are symmetric with respect to the origin (0,0). To proove analytically that a given function f is odd, we need to proove that:f(- x) = - f(x)## Example 1The formulas of the even functions f, g, h and i are given (see graphs above). Show analytically that each of these functions satisfies the property of an even function: f(-x) = f(x)Solution to Example 1
The formulae of the four functions f, g, h and i are given with the graphs of these functions. (see top graph above). 1) f(x) = x ^{2}
f(-x) = (- x) ^{2} = (-x)(-x) = x^{2}
From the above, f(-x) = f(x), hence f(x) is an even function. 2) g(x) = - | x | + 2 g(-x) = - | - x | + 2 The absolute value function function | x | is an even function, therefore | - x | = | x | we simplify g(-x) to g(-x) = - | x | + 2 g(-x) = g(x), hence g(x) is an even function. 3) h(x) = 10 e ^{- 0.2 x2}
h(- x) = 10 e ^{ - 0.2 (- x) 2} = 10 e^{ - 0.2 x 2}
Since h(-x) = h(x), h(x) is an even function. 4) i(x) = 8 cos(x) i(- x) = 8 cos( - x) The cosine function is an even function, therefore cos(-x) = cos(x) which gives i(- x) = 8 cos( - x) = 8 cos(x) i(-x) = i(x) and therefore i(x) ia an even function. ## Example 2The formulas of the odd functions j, k, l and m are given (see graphs above). Show analytically that each of these functions satisfies the property of an odd function: f(- x) = - f(x)Solution to Example 2
The formulae of the four functions j, k, l and m are given with the graphs of these functions. (see top graph above). 1) j(x) = x ^{3}
j(-x) = (- x) ^{3} = (-x)(-x)(-x) = - x^{3}
From the above, j(-x) = - j(x), hence j(x) is an odd function. 2) k(x) = - x k(-x) = - (-x) = x From the above, k(-x) = - k(x), hence k(x) is an odd function. 3) l(x) = 6 sin(x) l(-x) = 6 sin( - x) The sine function is an odd function, therefore sin(-x) = - sin(x) which gives l(-x) = 6 sin( - x) = - 6 sin(x) l(-x) = - l(x) and therefore l(x) ia an odd function. 3) m(x) = e ^{x} - e^{-x}
m(- x) = e ^{(- x)} - e^{-(-x)}
Simplify = e ^{- x} - e^{x}
and rewrite as = - ( - e ^{- x} + e^{x} )
m(-x) = - m(x) and therefore m(x) is an odd function. ## ExercisesVerify analytically whether each of these functions is even, odd or neither?1. f(x) = - 2 x ^{2} + 4
2. g(x) = 3 | x | - x ^{2}
3. h(x) = 1 / x 4. i(x) = tan(x) 5. j(x) = x ^{2} + 3 x
Detailed solutions to Above Exercises
1. f(x) = - 2 x ^{2} + 4
Let us find f(-x) = - 2 (-x) ^{2} + 4 = - 2 x^{2} + 4
f(-x) is equal to f(x) so function f is even. 2. g(x) = 3 | x | - x ^{2}
g(-x) = 3 | - x | - (-x) ^{2} = 3 | x | - x^{2} and is equal to g(x) hence function g is even.
3. h(x) = 1 / x h(-x) = 1 / (-x) = - 1 / x and is equal to - h(x), and therefore h is odd. 4. i(x) = tan(x) i(-x) = tan(-x) = - tan(x), function i is odd. 5. j(x) = x ^{2} + 3 x
j(-x) = (- x) ^{2} + 3 (- x) = x^{2} - 3 x
j(-x) is not equal to j(x) and j(-x) is not equal to - j(x). Therefore j(x) is neither an even nor an odd function. ## More References and LinksProperties of Trigonometric FunctionsApplications, Graphs, Domain and Range of Functions |