Helpp?
Graphs of functions !!
I need to determine if these are odd or even?
1. f(x)=x^5-x
2. f(x)=5
3.f(x)=x^4+2x^3
even if f(-x) = f(x)
odd if f(-x) = -f(x)
even if all powers are even
odd if all powers are odd
remember that 0 is an even power
So These Are all even?!
Sorry I'm still kind of lost..
come on. can't tell even and odd numbers?!?!?
x^5-x = x^5 - x^1
5 and 1 are both odd, so f(x) is odd
5 = 5x^0
0 is an even power, so f(x) is even
x^4 + 2x^3
4 is even, 3 is odd, so f(x) is neither even nor odd.
Sure, I can help you determine if these functions are odd or even.
To determine if a function is odd or even, we need to examine its symmetry properties.
1. f(x) = x^5 - x
To check if this function is odd or even, we need to check its symmetry with respect to the y-axis (even symmetry) and origin (odd symmetry).
For even symmetry, we need to verify f(x) = f(-x). Let's substitute -x for x in the equation:
f(-x) = (-x)^5 - (-x)
= -x^5 + x
Since -x^5 + x is not equal to x^5 - x, f(x) ≠ f(-x), and therefore, this function does not have even symmetry.
For odd symmetry, we need to verify f(x) = -f(-x). Let's substitute -x for x in the equation:
-f(-x) = -(-x)^5 + (-x)
= x^5 + x
Since x^5 + x is not equal to x^5 - x, f(x) ≠ -f(-x), and therefore, this function does not have odd symmetry.
Thus, the function f(x) = x^5 - x is neither odd nor even.
2. f(x) = 5
The function f(x) = 5 is a constant function. Constant functions are neither odd nor even because they do not exhibit any symmetry.
3. f(x) = x^4 + 2x^3
To determine if this function is odd or even, we'll follow the same process as before.
For even symmetry, we need to verify f(x) = f(-x). Let's substitute -x for x in the equation:
f(-x) = (-x)^4 + 2(-x)^3
= x^4 - 2x^3
Since x^4 - 2x^3 is not equal to x^4 + 2x^3, f(x) ≠ f(-x), and therefore, this function does not have even symmetry.
For odd symmetry, we need to verify f(x) = -f(-x). Let's substitute -x for x in the equation:
-f(-x) = -(x^4 + 2x^3)
= -x^4 - 2x^3
Since -x^4 - 2x^3 is equal to x^4 + 2x^3, f(x) = -f(-x), and therefore, this function has odd symmetry.
Hence, the function f(x) = x^4 + 2x^3 is odd.
I hope this clarification helps you understand how to determine if a function is odd or even by examining its symmetry properties. Let me know if there's anything else I can assist you with!