Question

Which statement best describes how to determine whether f(x) = x3 + 5x + 1 is an even function?
Determine whether –(x3 + 5x + 1) is equivalent to x3 + 5x + 1.
Determine whether (–x)3 + 5(–x) + 1 is equivalent to x3 + 5x + 1.
Determine whether –x3 + 5x + 1 is equivalent to –(x3 + 5x + 1).
Determine whether (–x)3 + 5(–x) + 1 is equivalent to –(x3 + 5x + 1).

Answer 1

To determine whether the function \( f(x) = x^3 + 5x + 1 \) is an even function, we need to check if \( f(-x) = f(x) \) for all \( x \).

1. **Calculate \( f(-x) \):**
\[
f(-x) = (-x)^3 + 5(-x) + 1 = -x^3 - 5x + 1
\]

2. **Check if \( f(-x) \) is equal to \( f(x) \):**
\[
f(x) = x^3 + 5x + 1
\]
Since \( f(-x) \neq f(x) \), the function is NOT even.

3. **Check if \( f(-x) \) equals \( -f(x) \):**
\[
-f(x) = -(x^3 + 5x + 1) = -x^3 - 5x - 1
\]
We can see that \( f(-x) \neq -f(x) \) as well.

Based on this analysis, the statement that best describes the process of determining whether \( f(x) \) is an even function is:

**"Determine whether \( (-x)^3 + 5(-x) + 1 \) is equivalent to \( x^3 + 5x + 1 \)."**

This checks whether \( f(-x) \) equals \( f(x) \), which is the definition of an even function.