How can we determine if the function f(x) = x^3 + 5x + 1 is an even function?

To determine whether the function f(x) = x³ + 5x + 1 is an even function, we need to apply the definition of an even function.

An even function is one that satisfies the condition f(-x) = f(x) for all values of x in its domain. In simpler terms, this means that when we plug in the negative of x into the function, the output should be the same as when we plug in x itself.

Let’s perform the calculation:

1. Start by calculating f(-x):

Substituting -x into the function, we get:

f(-x) = (-x)3 + 5(-x) + 1
     = -x3 - 5x + 1

2. Next, we will compare f(-x) with f(x):

We already know that:

f(x) = x3 + 5x + 1

3. Now let’s see if f(-x) = f(x):

Comparing the two results:

f(-x) = -x3 - 5x + 1   <>   f(x) = x3 + 5x + 1

Since f(-x) is not equal to f(x), we can conclude that:

f(x) = x³ + 5x + 1 is not an even function.

In general, for a function to be classified as even, its output must mirror its input around the y-axis. Functions that contain odd powers of x, like x³ in this case, typically do not satisfy the even function criteria. Therefore, we have verified that f(x) is indeed an odd function, as a further check would demonstrate that f(-x) = -f(x).