Handout 2: Odd and Even Functions

Note that this means that the graph of an even function is symmetric about x = 0 and the graph of an odd function is symmetric about the origin.

Whenever we want to check whether a function satisfies a property like even or odd then we check whether the definition of the property holds. So in this case to determine whether a function is even or odd we just have to calculate f(-x) and compare it with f(x).

Show that coshx is an even function and sinhx is an odd function.

Recall that coshx = [(e^x+e^-x)/2] and that sinhx = [(e^x-e^-x)/2]. Let f(x) = coshx. Then

f(-x) = cosh(-x) = e-x+ex 2 = ex+e-x 2 = coshx = f(x)

hence f is an even function. Now let g(x) = sinhx. Then

g(-x) = sinh(-x) = e-x-ex 2 = - ex-e-x 2 = -sinhx = -g(x)

hence g is an odd function.

Recall that sin is an odd function and cos is an even function.

Exercise: Sketch the graphs of sinhx, coshx, sinx and cosx to convince yourself of the symmetries involved in even / odd functions.

Show that f(x) = sinhx sinx + 2 coshx is an even function.

f(-x) = sinh(-x) sin(-x) + 2cosh(-x) = (-sinhx) (-sinx) + 2 coshx = sinhx sinx + 2 coshx = f(x)

and hence f is an even function.

Let f(x) be an odd function and g(x) be an even function. Let h(x) = (f(x))²+g(x). Show that h is an even function.

h(-x) = (f(-x))2 + g(-x) = (-f(x))2 + g(x) = (f(x))2 +g(x) = h(x)

Notice that in the second line we have used the fact that f is odd and g is even.