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

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).

GPT-4o mini · Answer

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.