Video: Even and Odd Functions

In this video, we will learn how to decide whether a function is even, odd, or neither both from a graph of the function and from its rule.

15:58

Video Transcript

In this lesson, we’re going to learn how to decide whether a function is even, odd, or neither, both by considering the graph of that function and from the equation itself. We’re going to begin, though, by looking at the definition of the parity of a function. Parity describes whether a function is even or odd. But what does it actually mean for that function to be even or odd? Well, to decide whether a function is odd or even or indeed neither, the first thing we do is we check the domain of the function. We need the domain to be centered at the origin. If we answer no to this question, then the function can be neither odd nor even.

For instance, let’s imagine the domain of our function is the open interval from negative eight to eight. 𝑥 equals zero is exactly halfway between these, and so the domain is indeed centered at 𝑥 equals zero. But what about the domain the left-closed right-open interval from negative four to two? Well, no. 𝑥 equals zero isn’t exactly halfway through here. And so a function with this domain would be neither odd nor even. However, if we’re able to answer yes to that earlier question, then we could say that the function is even if 𝑓 of negative 𝑥 is equal to 𝑓 of 𝑥. But it’s odd if 𝑓 of negative 𝑥 is equal to negative 𝑓 of 𝑥. And if the function 𝑓 of 𝑥 satisfies neither of these criteria, then it is neither odd nor even.

And so one technique we have to check whether a function is odd or even or indeed neither is to substitute negative 𝑥 into the function and see what we get. However, there is also a geometric approach. Let’s see where that approach comes from.

Determine whether the function represented by the following figure is even, odd, or neither even nor odd.

And then we have a graph of the function shown. So let’s recall how to check the parity of a function, how to check whether it’s even or odd. Well, the first thing we do is ask ourselves, is the domain of this function centered at 𝑥 equals zero? We might recall that the domain of a function is the set of possible inputs, the set of values of 𝑥, that we can substitute into the function. And we can read that domain from the graph.

Now we do need to be a little bit careful because the graph doesn’t actually appear to be defined at 𝑥 equals zero. In fact, the domain is the union of the left-closed right-open interval from negative eight to zero and the left-open right-closed interval from zero to eight. This is centered at 𝑥 equals zero. Zero is exactly halfway through this domain. And so we can now say yes to this question. And we are able to move on to the next part.

We can say that if 𝑓 of negative 𝑥 is equal to 𝑓 of 𝑥, the function is even, and it’s odd if 𝑓 of negative 𝑥 is equal to negative 𝑓 of 𝑥. Well, one way we can establish whether either of these is true is to choose a value of 𝑥. For instance, let’s use the point 𝑥 equals five. When 𝑥 is equal to five, the value of our function, the 𝑦-value, is negative one. So 𝑓 of five is negative one. And then this means that negative 𝑥 must be negative five. And so we need to read the 𝑦-value when 𝑥 is negative five. 𝑓 of negative five is also negative one. So it does look like this might be an even function. But let’s check with another value.

Let’s choose 𝑥 equals one. 𝑓 of one is roughly equal to negative 4.1. Then negative 𝑥 will be equal to negative one. And once again, 𝑓 of negative one is roughly equal to negative 4.1. And so for the two values we’ve tried, 𝑓 of negative 𝑥 is equal to 𝑓 of 𝑥. But actually, if we look carefully, we see that that function itself has reflectional symmetry about the 𝑦-axis. And so indeed, every value of 𝑓 of 𝑥 must be equal to every value of 𝑓 of negative 𝑥. And so we can say that the function itself must be even.

In fact, we can generalize this. We can say that an even function has reflectional symmetry about the 𝑦-axis. Take, for instance, the graph of 𝑓 of 𝑥 equals cos of 𝑥. This is completely symmetrical about the 𝑦-axis, and so cos of 𝑥 must be an even function. This is a general result that we can quote. The same cannot be said for an odd function, though. An odd function does have symmetry, but it has rotational symmetry about the origin order two. In other words, it’s unchanged when it’s rotated by 180 degrees about the origin. Take, for instance, the function 𝑓 of 𝑥 equals sin 𝑥. If we rotate the graph of this function 180 degrees about the origin, it will end up looking exactly the same as it did originally.

The same can be said for the tangent function. Despite its asymptotes, if we rotate this 180 degrees about the origin, about zero, zero, it will look exactly the same, as long as, of course, its domain is also symmetrical. Its domain must be centered at 𝑥 equals zero. Now that we have these graphical representations, let’s have a look at another example.

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.

In our next example, we’ll look at how to decide whether a function is even or odd given the equation.

Is the function 𝑓 of 𝑥 equals 𝑥 to the fifth power times tan of six 𝑥 to the fourth power even, odd, or neither even nor odd?

Let’s recall how we check the parity of a function. The first thing we do is check the domain of the function. We need that to be centered at 𝑥 equals zero. Then, if the answer is no, we can say that the function is neither even nor odd without performing any further tests. If the answer is yes, though, we say that it will be even if it satisfies 𝑓 of negative 𝑥 equals 𝑓 of 𝑥. And it will be odd if it satisfies 𝑓 of negative 𝑥 equals negative 𝑓 of 𝑥. Then, of course, if it satisfies neither of these, it will be neither even nor odd.

So let’s think about the domain of our function. Our function is the product of two functions. It’s the product of 𝑥 to the fifth power and tan of six 𝑥 to the fourth power. And so the domain of 𝑓 of 𝑥 will be the intersection of the domains of the respective parts of the function. Well, 𝑥 to the fifth power is a polynomial, so its domain is the set of real numbers or the open interval from negative ∞ to ∞. But what about the domain of the trigonometric part? Well, it’s all real numbers, except those that make cos of six 𝑥 equal to zero. But since the values of 𝑥 that make cos of six 𝑥 equal to zero are symmetrical about the 𝑦-axis, then we can say that the domain of tan of six 𝑥 to the fourth power must be centered at 𝑥 equals zero.

Since both domains are centered at 𝑥 equals zero, then we can answer yes to this first question, and we’re able to move on. We now see that it’s even if 𝑓 of negative 𝑥 is equal to 𝑓 of 𝑥 and odd if it’s equal to negative 𝑓 of 𝑥. And so let’s evaluate 𝑓 of negative 𝑥. To do so, we replace each instance of 𝑥 in our original function with negative 𝑥. And we get 𝑓 of negative 𝑥 is negative 𝑥 to the fifth power times tan of negative six 𝑥 to the fourth power. We’ll evaluate each part in turn. Let’s begin with negative 𝑥 to the fifth power. Since the exponent is odd, when we multiply this out, we’re going to get a negative result. Negative 𝑥 to the fifth power is as shown.

But what about the tan function? Well, we can actually quote the result that tan of 𝑥 is odd, meaning that tan of negative 𝑥 is equal to negative tan of 𝑥 and, in turn, the tan of negative six 𝑥 is equal to negative tan of six 𝑥. But of course, we’re raising this to the fourth power. We’re raising it to an even exponent. And we know when we raise a negative number to an even exponent, the result is positive. And so tan of negative six 𝑥 to the fourth power is just tan of six 𝑥 to the fourth power.

And so 𝑓 of negative 𝑥 is therefore equal to negative 𝑥 to the fifth power times tan of six 𝑥 to the fourth power. So does this satisfy either of our criteria, is it even or odd? Well, yes. If we look at it carefully, we see it’s the same as negative 𝑓 of 𝑥. 𝑓 of negative 𝑥 is equal to negative 𝑓 of 𝑥. And so the function must be odd.

Let’s now consider how this process might work, given a piecewise function.

Determine whether the function 𝑓 is even, odd, or neither, given that 𝑓 of 𝑥 is equal to negative nine 𝑥 minus eight if 𝑥 is less than zero and nine 𝑥 minus eight if 𝑥 is greater than or equal to zero.

Let’s recall the steps that allow us to check the parity of a function, in other words, whether it’s even or odd. Firstly, we establish whether the domain is centered at 𝑥 equals zero. If the answer to this is yes, then we can say that it is even if 𝑓 of negative 𝑥 equals 𝑓 of 𝑥 and odd if 𝑓 of negative 𝑥 equals negative 𝑓 of 𝑥. Well, we see by looking at the values here that the domain of our function is simply the set of real numbers. And of course, that set is very obviously centered at 𝑥 equals zero. And so we answer yes to the first question. And so now we’re going to consider the second criteria.

Since this is a piecewise function, we need to be really careful. When 𝑥 is less than zero, we’re working with 𝑓 of 𝑥 equals negative nine 𝑥 minus eight. And when it’s greater than or equal to zero, we’re working with nine 𝑥 minus eight. And so let’s consider both parts of our function individually. Let’s take negative nine 𝑥 minus eight. And we’re now going to find 𝑓 of negative 𝑥. It will be negative nine times negative 𝑥 minus eight. But of course, a negative multiplied by a negative is a positive. So 𝑓 of negative 𝑥 is equal to nine 𝑥 minus eight.

Notice that that is equal to the second part of our function. And it’s absolutely fine that it’s equal to the other part, since we’ve changed the sign of the value of 𝑥. And so we’re moving on to the other side of the piecewise function. And so for this part, yes, 𝑓 of negative 𝑥 is equal to 𝑓 of 𝑥. So that part certainly appears to be even. But let’s check the second part. Let’s take 𝑓 of 𝑥 equals nine 𝑥 minus eight and then find 𝑓 of negative 𝑥. It’s nine times negative 𝑥 minus eight, which is negative nine 𝑥 minus eight. We’ve changed the sign of the value of 𝑥, and we’ve ended up with 𝑓 of 𝑥 when 𝑥 is less than zero. And so this part of the function is also even. We can say that 𝑓 of negative 𝑥 is equal to 𝑓 of 𝑥. And we can therefore say that our piecewise function is even.

In our final example, we’ll look at how the domain can affect the parity of a function.

Determine whether the function 𝑓 of 𝑥 equals nine 𝑥 cubed is even, odd, or neither even nor odd, given that 𝑓 maps numbers from the left-open right-closed interval from negative seven to seven onto the set of real numbers.

Remember, to check the parity of a function, we first check whether its domain is centered at 𝑥 equals zero. If not, then the function will be neither even nor odd. But if it is, we can say it’s even if it satisfies 𝑓 of negative 𝑥 equals 𝑓 of 𝑥. And it’s said to be odd if it satisfies 𝑓 of negative 𝑥 equals negative 𝑓 of 𝑥. Now the domain of our function is the set of possible inputs. It’s the left-open right-closed interval from negative seven to seven. And at first glance, it does look like our domain is centered at the origin. But notice that we have a curly or round bracket here and a square bracket here. This means that our domain doesn’t include the value of 𝑥 equals negative seven, but it does include the value of 𝑥 equals seven. And so it’s kind of lopsided. Its center will be slightly to the right of zero. And so we can say that this function is in fact neither even nor odd.

Notice that had we failed to check this, we might have concluded that the function is odd. And this is because 𝑓 of negative 𝑥 here is nine times negative 𝑥 cubed. But negative 𝑥 cubed is negative 𝑥 cubed. And so 𝑓 of negative 𝑥 is the same as negative nine 𝑥 cubed, which is the same as negative 𝑓 of 𝑥. And so this shows us how important it is that we do check the domain. You might also like to think about this graphically. A function is odd if it has rotational symmetry order two about the origin. In other words, if we rotate it 180 degrees about zero, zero, it will look exactly the same. This cannot be the case here because when we rotate it, we take the point that we’re including and map it onto the point that we’re not including and of course vice versa. And so it won’t look exactly the same. And so we determine that our function under these restraints is neither even nor odd.

We’ll now clarify the key points from this lesson. Parity describes whether a function is even or odd. And we recall that, to check the parity, we first check whether the domain of the function is centered at 𝑥 equals zero. If the answer to this is no, then the function can be neither even nor odd. But if the answer is yes, we say that the function is even if it satisfies 𝑓 of negative 𝑥 equals 𝑓 of 𝑥 and it’s odd if it satisfies 𝑓 of negative 𝑥 equals negative 𝑓 of 𝑥. Once again, if it satisfies neither of these, it can be neither even nor odd.

We saw that we can also use the graph of the functions to determine whether they are even or odd. An even function will have a reflectional symmetry about the 𝑦-axis, whereas odd functions have rotational symmetry order two about the origin. They’ll be unchanged when they’re rotated by 180 degrees about zero, zero.

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