To determine whether the function f(x) = x^4 + x^3 is an even function, we need to check the defining property of even functions.
An even function is one that satisfies the condition:
f(-x) = f(x) for all x in the domain of f.
This means that if we replace x with -x in the function, we should receive the same result as the original function.
Now, let’s apply this to our function:
First, we calculate f(-x):
f(-x) = (-x)^4 + (-x)^3
Calculating each term:
- (-x)^4 = x^4 (since raising to an even power negates the negative sign),
- (-x)^3 = -x^3 (since raising to an odd power preserves the negative sign).
Now, substituting these back into our function gives:
f(-x) = x^4 – x^3
Next, we compare f(-x) to f(x):
f(x) = x^4 + x^3
Since:
f(-x) = x^4 – x^3 is not equal to f(x) = x^4 + x^3
we can conclude that f(x) = x^4 + x^3 does not satisfy the even function property.
In contrast, an even function would yield identical results when substituting x with -x throughout its domain.
Thus, the statement that best describes how to determine whether f(x) = x^4 + x^3 is an even function is that it is NOT an even function because f(-x) ≠ f(x).