Determine if the function is even, odd or neither.
f(x)=2x^5+2x^3
f^-1x=2(-x)^5+2(-x)^3
f^-1x=-2x^5+-2x^3
f^-1x=-2x^5-2x^3
f(x)=2x^5+2x^3
f^-1x=2(-x)^5+2(-x)^3
f^-1x=-2x^5+2(-x)^3
f^-1x=(-2x^5)+(-2x^3)
f^-1x=-2x^5-2x^3
Not even. This function is odd.
that is f(-x) you want but your answer is correct.
f(-x) = 2(-x)^5 + 2(-x)^3
but (-x)^5 = - x^5
and (-x)^3 = - x^3
so
f(-x) = -2 x^5 - 2 x^3
= -f(x)
so odd because f(-x) = - f(x))
To determine if a function is even, odd, or neither, we need to check if the function satisfies certain properties when we substitute -x for x.
An even function is one that satisfies f(x) = f(-x) for all x in the domain. If f(x) is even, the graph of the function is symmetric about the y-axis.
An odd function is one that satisfies f(x) = -f(-x) for all x in the domain. If f(x) is odd, the graph of the function is symmetric about the origin.
Let's check the given function:
f(x) = 2x^5 + 2x^3
Now, substitute -x for x:
f(-x) = 2(-x)^5 + 2(-x)^3
f(-x) = -2x^5 + -2x^3
Comparing f(x) and f(-x), we can see that f(x) ≠ f(-x) and f(x) ≠ -f(-x). Therefore, the function does not satisfy the properties of an even or odd function. Thus, the function is neither even nor odd.