determine odd and even functions

Question

Hi,

 

Need some help determining if the following functions are odd even or neither.

 

f(x)=4/g(x)

 

f(x)=g(cosx)

 

f(x)=e^-g(x)

 

 

Thanks in advance

 

Michael

Answer 1

Hi Michael.

 

Not sure what g(x) is in your problem. I'm sure it's defined somewhere in your text.

 

Anyways, odd and even functions have the following properties:

 

Even functions:

 

f(-x) = f(x)

 

Odd functions:

 

f(-x) = -f(x)

 

**

 

For example, with even functions, take a look at f(x) = x^2

 

This is an even function, because

 

f(-x) = (-x)^2 = x^2 = f(x)

 

So f(-x) = f(x)

 

Or take a look at f(x) = cos(x)

 

This is also an even function, because

 

f(-x) = cos(-x) = cos(x) = f(x)

 

So f(-x) = f(x)

 

You can verify this with real numbers if you'd like.

 

**

 

Now with odd functions, take a look at g(x) = x^3

 

This is an odd function because

 

g(-x) = (-x)^3 = -x^3 = -g(x)

 

So g(-x) = -g(x)

 

Or consider g(x) = sin(x)

 

This is also an odd function because

 

g(-x) = sin(-x) = -sin(x) = -g(x)

 

So g(-x) = -g(x)

 

Again, you can verify with real numbers if you'd like.

 

With all of your problems, plug in -x and see if the value your function returns is f(x) or -f(x). If it returns f(x) it's even. If it returns -f(x) it's odd.

 

Hope this helps.