Determine whether each function is odd, even or neither.
a) f(x) = -6x^5+2x^3
b) f(x) = x^4-x^2+1
Polynomials with all terms in even degree are even functions, and those with all odd degree terms are odd functions.
For this purpose, a constant k can be considered as k*x^0, so it is even.
A mix of even and odd degree terms is neithter odd nor even.
Example:
f(x)=x^6+x^2+4 is even
f(x)=x^5+x^3+6x is odd
f(x)=x^6+x+1 is neither odd nor even
A test can be applied, which is also the definition of even and odd functions:
if f(-x)=f(x) then f(x)is an even function
if -f(-x)=f(x) then f(x) is an odd function
Even
To determine whether a function is odd, even, or neither, we need to consider the symmetry of the function.
a) f(x) = -6x^5 + 2x^3
To check if this function is even, we need to verify if f(x) = f(-x) for all x.
Let's substitute -x for x in the given function:
f(-x) = -6(-x)^5 + 2(-x)^3
= -6(-x)(-x)(-x)(-x)(-x) + 2(-x)(-x)(-x)
= -6x^5 + 2x^3
Since f(-x) = f(x), this function is even.
b) f(x) = x^4 - x^2 + 1
To check if this function is odd, we need to verify if f(x) = -f(-x) for all x.
Let's substitute -x for x in the given function and check if f(x) = -f(-x):
f(-x) = (-x)^4 - (-x)^2 + 1
= x^4 - x^2 + 1
Since f(-x) is not equal to -f(x), this function is neither odd nor even.
To determine whether a function is odd, even, or neither, we need to examine how the function behaves when input values are replaced with their negatives.
Let's start with function a) f(x) = -6x^5 + 2x^3.
1. Odd Function:
A function f(x) is odd if f(-x) = -f(x) for all x in the domain.
To test if the function is odd, we substitute -x for x in the function and check if the result is equal to the negative of the original function:
f(-x) = -6(-x)^5 + 2(-x)^3
= -6(-x^5) + 2(-x^3)
= -6(-x^5) + 2(-x^3)
= 6x^5 - 2x^3
Since f(-x) = 6x^5 - 2x^3, and -f(x) = -(-6x^5 + 2x^3) = 6x^5 - 2x^3, we can see that f(-x) = -f(x) for all x in the domain.
Therefore, function a) f(x) = -6x^5 + 2x^3 is an odd function.
2. Even Function:
An even function f(x) is even if f(-x) = f(x) for all x in the domain.
Since f(-x) = 6x^5 - 2x^3, we can see that f(-x) is not equal to f(x) = -6x^5 + 2x^3 for all x in the domain.
Therefore, function a) f(x) = -6x^5 + 2x^3 is not an even function.
Next, let's move on to function b) f(x) = x^4 - x^2 + 1.
1. Odd Function:
To test if the function is odd, we substitute -x for x in the function and check if the result is equal to the negative of the original function:
f(-x) = (-x)^4 - (-x)^2 + 1
= x^4 - x^2 + 1
Since f(-x) = x^4 - x^2 + 1 and -f(x) = -(x^4 - x^2 + 1) = -x^4 + x^2 - 1, we can see that f(-x) is not equal to -f(x) for all x in the domain.
Therefore, function b) f(x) = x^4 - x^2 + 1 is not an odd function.
2. Even Function:
To test if the function is even, we substitute -x for x in the function and check if the result is equal to the original function:
f(-x) = (-x)^4 - (-x)^2 + 1
= x^4 - x^2 + 1
Since f(-x) = x^4 - x^2 + 1 = f(x), we can see that f(-x) = f(x) for all x in the domain.
Therefore, function b) f(x) = x^4 - x^2 + 1 is an even function.
In summary:
a) f(x) = -6x^5 + 2x^3 is an odd function.
b) f(x) = x^4 - x^2 + 1 is an even function.