Before we begin, let's recall two important definitions.
- A function f has when f(-x)=f(x) for all x in its domain.
- A function f has when f(-x)=-f(x) for all x in its domain.
Let's see how the graphs of these types of functions look.
Even functions
Odd functions
Reset
Consider the graphs above. We can see that for even symmetry, if
(x,y) is on the graph, then
(-x,y) is also on the graph. Meanwhile, for an odd symmetry, if
(x,y) is on the graph, then
(-x,-y) is also on the graph. Now, consider the given function.
s(x)=-x3+2x−9
Let's calculate
s(-x).
s(-x)=-(-x)3+2(-x)−9 s(-x)=-(-x3)+2(-x)−9 s(-x)=x3+2(-x)−9
s(-x)=x3−2x−9
Next, let's calculate
-s(x).
-s(x)=-(-x3+2x−9) -s(x)=x3−2x+9
Finally, let's think about what these results tell us.
s(x)
|
s(-x)
|
-s(x)
|
-x3+2x−9
|
x3−2x−9
|
x3−2x+9
|
Since s(x)=s(-x) and s(-x)=-s(x), the function s has neither an even nor an odd symmetry.