Question Video: Determining If a Graphed Function Is Even, Odd, or Neither Mathematics

Is the function represented by the figure even, odd, or neither even nor odd?

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Video Transcript

Is the function represented by the figure even, odd, or neither even nor odd?

Remember, we can check the parity of a function by considering its graph. Before we do, though, we do need to decide whether the domain is centered at 𝑥 equals zero. If the answer is yes, then we move on to the next stage. But if it’s not, the function cannot be even nor odd. So let’s find the domain of our function. We remember that the domain is the set of possible inputs to the function, the set of values of 𝑥 that go into the function. The smallest value of 𝑥 that is in our function is 𝑥 equals two, and the largest possible value of 𝑥 is 𝑥 equals six. And so we see that the domain is the closed interval from two to six. This domain is actually centered at 𝑥 equals four. The halfway point is 𝑥 equals four. And so actually, we’re answering no to this first question. And so the function is neither even nor odd.

But let’s have a look at a common misconception here. When we think about the graphical representation of functions, we know that these functions are even if they have reflectional symmetry about the 𝑦-axis. And they’re odd if they have rotational symmetry order two about the origin. Now, our graph does appear to have some rotational symmetry. If we take the center to be four, one, then it does indeed have rotational symmetry order two. If I rotate that graph 180 degrees, it will look exactly the same. But this is not about the origin. Its center is four, one. And so that confirms to us that this graph is neither even nor odd.

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