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### Course: Algebra 2 > Unit 9

Lesson 3: Symmetry of functions- Function symmetry introduction
- Function symmetry introduction
- Even and odd functions: Graphs
- Even and odd functions: Tables
- Even and odd functions: Graphs and tables
- Even and odd functions: Equations
- Even and odd functions: Find the mistake
- Even & odd functions: Equations
- Symmetry of polynomials

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# Even and odd functions: Tables

Even functions are symmetrical about the y-axis: f(x)=f(-x). Odd functions are symmetrical about the x- and y-axis: f(x)=-f(-x). Let's use these definitions to determine if a function given as a table is even, odd, or neither.

## Want to join the conversation?

- How many points we should examine to be sure if a function is odd or even ?(13 votes)
- In the questions with the table you should just check every value given.

On graphs you can eyeball it.

If you're just given a function you input a -x and see what happens.

So for example you have f(x) = 4x^2 + 3

f(-x) = 4(-x)^2 + 3 = 4(x)^2 + 3 = f(x)

which means the function is even.

On the other hand g(x) = 3x + 2

g(-x) = 3(-x) + 2 = -3x + 2

so g(x) is not even, because f(-x) != f(x).

For odd functions it works the same.

h(x) = x^3 - x

h(-x) = (-x)^3 - (-x) = -x^3 + x = - (x^3 - x) = -h(x)

So h(x) is odd.(17 votes)

- I always visualize it in my mind. Also sobe subjects might take time to comprehend, try to be patient, use different sources.(6 votes)

- How can I find out if the graph is even or odd when there aren't any points that line up?(2 votes)
- how can you tell between an even or a odd function can someone please help me?(1 vote)
- should be the leading factor's exponents, see if it is an even of a odd.(3 votes)

- in problem one, dont even functions have to go through the origin? It says that the y intercept is (0,1)(1 vote)
- No, only odd functions have to pass through the origin. Even functions are function that are symmetric about the y-axis, i.e. left side of y-axis = right side of y-axis.(2 votes)

- why are odd functions symitrical(1 vote)
- how to determinea functionif itis odd,even or neither by obseerving a table(0 votes)

## Video transcript

- [Instructor] We're told
this table defines function f. All right. For every x, they give us
the corresponding f of x. According to the table, is
f even, odd, or neither? So pause this video and see if you can figure that out on your own. All right, now let's
work on this together. So let's just remind
ourselves the definition of even and odd. One definition that we can
think of is that f of x, if f of x is equal to f of negative x, then we're dealing with an even function. And if f of x is equal to the negative of f of negative x, or another way of saying
that, if f of negative x. If f of negative x, instead
of it being equal to f of x, it's equal to negative f of x. These last two are equivalent. Then in these situations, we are dealing with an odd function. And if neither of these are true, then we're dealing with neither. So what about what's going on over here? So let's see. F of negative seven is
equal to negative one. What about f of the
negative of negative seven? Well, that would be f of seven. And we see f of seven here is
also equal to negative one. So at least in that case and that case, if we think of x as seven, f of x is equal to f of negative x. So it works for that. It also works for
negative three and three. F of three is equal to
f of negative three. They're both equal to two. And you can see and you can
kind of visualize in your head that we have symmetry around the y-axis. And so this looks like an even function. So I will circle that in. Let's do another example. So here, once again, the
table defines function f. It's a different function f. Is this function even, odd, or neither? So pause this video and
try to think about it. All right, so let's
just try a few examples. So here we have f of five is equal to two. F of five is equal to two. What is f of negative five? F of negative five. Not only is it not equal to two, it would have to be equal to two if this was an even function. And it would be equal to negative two if this was an odd
function, but it's neither. So we very clearly see just
looking at that data point that this can neither be even, nor odd. So I would say neither or
neither right over here. Let's do one more example. Once again, the table defines function f. According to the table, is
it even, odd, or neither? Pause the video again. Try to answer it. All right, so actually
let's just start over here. So we have f of four is
equal to negative eight. What is f of negative four? And the whole idea here is I wanna say, okay, if f of x is equal to something, what is f of negative x? Well, they luckily give
us f of negative four. It is equal to eight. So it looks like it's not equal to f of x. It's equal to the negative of f of x. This is equal to the
negative of f of four. So on that data point alone, at least that data point
satisfies it being odd. It's equal to the negative of f of x. But now let's try the other
points just to make sure. So f of one is equal to five. What is f of negative one? Well, it is equal to negative five. Once again, f of negative x is equal to the negative of f of x. So that checks out. And then f of zero, well, f of zero is of
course equal to zero. But of course if you say what is the negative of f of, if you say what f of negative of zero, well, that's still f of zero. And then if you were to
take the negative of zero, that's still zero. So you could view this. This is consistent still with being odd. This you could view as the
negative of f of negative zero, which of course is still going to be zero. So this one is looking
pretty good that it is odd.