Even and Odd Functions

 

 

For a function to be of Even Degree it must meet the following:

 

 

For a function to be of Odd Degree it must meet the following:

 

 

 

Examples:

 

1. f(x) = 3x^4 + 2x^2

 

Since each term of f(x) has an even exponent , f(x) is an even function. As well, f(x) has line symmetry on the y-axis. The last thing to do is confirm that f(-x) = f(x).

 

f(-x) = 3(-x)^4 + 2(-x)^2

f(-x) = 3x^4 + 2x^2

f(-x) = f(x)

 

Above, to check that the function is even, sub in –x for x and then simplify. When you simplify, if f(-x) = f(x)  (f(x) being the original equation), then the function is even.

 

Since f(-x) = f(x) , this function is even.

 

2. f(x) = 2x^3 – 4x

 

Since each term of f(x) has an odd exponent, it is odd. As well this function has point symmetry on a graph around the origin. The last thing to do is confirm that f(-x) = -f(x).

 

f(-x) = 2(-x)^3 -4(-x)

f(-x) = -2x^3 + 4x

f(-x) = -(2x^3 - 4x)

 

Above, in the check when the negative is divided out, the function is in it’s original form with a negative outside of the brackets (-f(x)). That means that the function is odd because f(-x) = -f(x).

Since f(-x) = -f(x) , this function is odd.

 

3. f(x) = 4x^4 – 3x^3 + 2x

 

This function would not be even or odd because some of the exponents are odd and some are even. To confirm, apply f(-x) and make sure it doesn’t result in f(x) or –f(x).

 

f(-x) = 4(-x)^4 – 3(-x)^3 + 2(-x)

f(-x) = 4(x)^4 + 3(x)^3 - 2x  -- At this point f(-x) does not equal f(x)

f(-x) = - (-4x^4 – 3x^3 + 2x) – At this point f(-x) does not equal –f(x)

 

Since f(-x) doesn’t equal f(x) or –f(x), therefore the function is nor positive or negative.

 

 

 

 

 

If you have any questions, leave them in the comments below.