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• >> This is Teresa Adams,

• and what we're doing today is finding the domain

• of a function.

• I'm going to look at the very basic function f

• of x equals x plus 1.

• This is an equation of a line.

• What we want to do when we're finding the domain

• of a function is find the value of x that we can put

• into our function so that the output will be real numbers.

• So as I look at the equation on this line,

• I have a line that looks like this.

• I'm going to cross at positive one, and I'm going

• to have a one, one slope.

• [ Pause ]

• And as I look at the graph of the function,

• I realize that no matter what value I put in for x

• that every output that I have for every y value

• that I have will be ok values.

• They'll all be real values.

• So for this function, my domain is going to be equal to x

• such that x is a real number.

• This is called set builder notation.

• If you don't like set builder notation, you can write it

• with integral notation.

• It runs from negative infinity to infinity.

• So a real basic function like this, we don't have to worry

• about what values of x that will be undefined

• or won't be a real number, then I can say that it's going

• to all be real numbers.

• If my function, say, is f of x is equal to the square root

• of x, then if I were to graph this function

• [ Pause ]

• it would look like something like that.

• So as we can see from the graph that I don't get

• to have any negative x values over here, and, in fact,

• if I were to put a negative x value in here, a negative one

• or negative two or something like that,

• you'll find that you have a complex number,

• and for right now, we want to stay in the real numbers.

• So we know that the domain of this one is x,

• such that x is greater than or equal to zero

• because you can take the square root of zero.

• Square root of zero is right there,

• and x belongs to the real numbers.

• Again, this is set builder notation,

• or if you'd like to do an integral notation,

• you have the domain is equal to zero to infinity, but you do get

• to include zero, so it's a squared off bracket all the way

• up to infinity.

• This excludes any values that are less than zero.

• Let's look at a more complex rational function.

• I have x over x minus one.

• Now, if I put in zero, I'm going to be fine.

• If I put in two, I'm going to be fine.

• If I put in negative two, I'm going to be fine.

• If I put in one, however,

• [ Pause ]

• I'm going to be undefined.

• So I don't really care what's going on on the top.

• I can put in any number I wanted on the top, but what I do want

• to prevent is I want to find prevent the denominator

• from being zero.

• So what I want to do is I want

• to find the restrictions only on the denominator.

• So since the denominator x minus one not equal to zero,

• and then I solve for x. So I have x minus one can't be zero.

• Adding one to both sides that means x cannot be one.

• This is my restriction.

• I build my domain for my restrictions.

• [ Pause ]

• So what I've got is my domain is equal to x

• such that x cannot equal one, and x is a real number.

• So that's the domain for that rational function.

• Let's look at another function.

• We have x minus two over x squared minus 4x minus 5.

• Now right now I've got to decide what values

• in the denominator would make this to be zero

• because that's what x cannot be.

• Those are my restrictions.

• At this point, I can't always clearly see what it is,

• so I have to factor this.

• So I know that this factors into x minus five multiplied

• by x minus one, x plus one.

• Now, looking at this, I know that this factor

• of the denominator cannot be zero,

• and I know that this factor of the denominator cannot be zero.

• These will give me my restrictions.

• X minus five cannot be zero, and x plus 1 cannot be zero.

• So x cannot be 5, and x cannot be negative one.

• These are my restrictions.

• [ Pause ]

• and from my restrictions, I build my domain.

• My domain is equal to x such that x is not equal

• to five nor negative one, but it is a real number.

• Again, I don't care about what's happening in the numerator,

• because a numerator I don't care if it's zero or not.

• I only want to make sure

• that the denominator is not equal to zero.

• What I don't want is the denominator equal to zero.

• I just have to prevent zero from being in the denominator