By comparing the values f(x) and f(-x), determine whether the function is (a) even or (b) odd. Use a graphing calculator to check your answer.
f(x) = 6
Answer
Since f(x) is 6 for every value of x, both f(x) and f(-x) are 6. This function is even. The graph confirms this:
Since 6≠ -6, this function is not odd.
Example 2
By comparing the values of f(x) and f(-x), determine whether thefunction is (a) even or (b) odd. Use a graphing calculator to check your answer.
f(x) = -x
Answer
We have f(x) = -x and f(-x) = -(-x) = x. These values are not the same, so f(x) is not even. These values are negatives of each other, though, so we can say f(x) is odd.
Example 3
By comparing the values f(x) and f(-x), determine whether each function is (a) even or (b) odd. Use a graphing calculator to check your answer.
f(x) = x3 + x4
Answer
f(x) = x3 + x4andf(-x) = (-x)3 + (-x)4 = -x3 + x4. Since f(x) are f(-x) are neither the same, nor negatives of each other, this function is neither even nor odd.
From the graph it looks like f(x) is trying to be even but failing.
Make sure to notice the bit around the origin where f(x) dips below the y-axis.
Example 4
By comparing the values f(x) and f(-x), determine whether the function is (a) even or (b) odd. Use a graphing calculator to check your answer.
f(x) = 0
Answer
We have f(x) = 0 and f(-x) = 0. These values are the same, so f(x) is even. These values are also negatives of each other, since 0 = -0. Therefore f(x) is also odd. Zero is such a tricky number.
