Before we begin, let's recall two important definitions.
- A function f is an when f(-x)=f(x) for all x in its .
- A function f is an when f(-x)=-f(x) for all x in its domain.
Let's see how the graphs of these types of functions look.
Consider the graphs above. We can see that for even functions, if
(x,y) is on the graph, then
(-x,y) is also on the graph. Meanwhile, for an odd function, if
(x,y) is on the graph, then
(-x,-y) is also on the graph. Now, consider the given function.
h(x)=x5+3x4
Let's calculate
h(-x).
h(-x)=(-x)5+3(-x)4
h(-x)=-x5+3(-x)4
h(-x)=-x5+3x4
Next, let's calculate
-h(x).
-h(x)=-(x5+3x4)
-h(x)=-x5−3x4
Finally, let's think about what these results tell us.
h(x)
|
h(-x)
|
-h(x)
|
x5+3x4
|
-x5+3x4
|
-x5−3x4
|
Since h(x)=h(-x) and h(-x)=-h(x), the function h is neither an even nor an odd function.