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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

>> This is Teresa Adams,

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A2 BEG US denominator domain function equal notation negative

Finding Domain of Functions

  • 576 1
    sf.cheng   posted on 2015/04/17
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