Final answer:
The function f(x)=7x⁴+5x+2 is neither even nor odd. It fails the even function test because f(-x) does not equal f(x), and it fails the odd function test because f(-x) is not the negative of f(x).
Step-by-step explanation:
To determine whether the function f(x)=7x⁴+5x+2 is even, odd, or neither, we must analyze it algebraically. For a function to be even, f(x) must equal f(-x), and for a function to be odd, f(-x) should equal -f(x). We can test this by substituting -x into the given function.
First, let's check if it's an even function:
f(-x) = 7(-x)⁴ + 5(-x) + 2 = 7x⁴ - 5x + 2.
This is not equal to f(x), because the middle term changes sign. Therefore, it is not an even function.
Now let's check if it's an odd function:
Since f(x) and f(-x) are not negatives of each other (there's a positive 7x⁴ term in both cases and the constant term +2 does not change sign), we can conclude that it is not an odd function either.
The function f(x)=7x⁴ + 5x + 2 is neither even nor odd because it does not meet the criteria for symmetry associated with even or odd functions.