Determine whether the function 𝑓 of 𝑥 equals nine 𝑥 cubed is even, odd, or neither even or odd, given that 𝑓 is defined for values of 𝑥 greater than negative seven and less than seven and 𝑥 is a real number.
Remember, a function is defined to be even if 𝑓 of negative 𝑥 is equal to 𝑓 of 𝑥 for all values of 𝑥 in the domain of 𝑓 of 𝑥. And it’s said to be odd if 𝑓 of negative 𝑥 is equal to negative 𝑓 of 𝑥. Of course, a function is neither odd nor even if neither of these definitions hold. In our question, we see that 𝑓 of 𝑥 is equal to nine 𝑥 cubed. So what’s 𝑓 of negative 𝑥? Well, if we replace 𝑥 with negative 𝑥, we see 𝑓 of negative 𝑥 is nine times negative 𝑥 cubed. Well, negative 𝑥 cubed is negative 𝑥 cubed. So 𝑓 of negative 𝑥 is negative nine 𝑥 cubed.
And so far, so good. 𝑓 of negative 𝑥 does indeed look like it’s equal to negative 𝑓 of 𝑥, making this an odd function. However, we’re going to need to be a little bit careful with the domain of our function. We see that 𝑥 takes values greater than negative seven and less than or equal to seven. So we evaluate 𝑓 of seven. It’s nine times seven cubed, which is 3087. But negative 𝑓 of negative seven is not equal to 𝑓 of seven. And this is because 𝑓 of negative seven is not defined on the domain of 𝑓. 𝑥 equals negative seven is outside the domain of our function.
So we’ve just shown that 𝑓 of negative 𝑥 is not equal to negative 𝑓 of 𝑥 for all values of 𝑥 in the domain of our function and nor is 𝑓 of negative 𝑥 equal to 𝑓 of 𝑥 . This means the function is neither odd nor even.