WEBVTT
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here we have functions. F N g. So
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let's start by finding F plus g. So we're
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going to add these functions together. X cubed plus
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two X squared plus three X squared minus one.
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We can add the like terms and we get X
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cubed plus five X squared minus one Breath plus G
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for F minus. G will subtract them. So
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we have X cubed plus two X squared, minus
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the quantity three X squared minus one. And so
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let's distribute the negative sign to have X cubed plus
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two X squared minus three X squared, plus one
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. And then let's combine like terms and we have
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X cubed minus X squared plus one and thats f
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minus g. Now let's look at the domains so
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F and G or both polynomial functions and polynomial functions
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always have a domain of all real numbers, which
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we can write as negative infinity to infinity. And
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when you add or subtract functions, the domain of
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the new function will be just the intersection of the
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two separate functions, and the intersection of all real
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numbers and all real numbers is all real numbers.
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All right, now, let's take a look at
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the product and the quotient. So if we multiply
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f and G X cubed plus two x squared multiplied
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by three X squared minus one, we might want
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to multiply that out so we'll use the foil process
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will multiply the first and we get three X to
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the fifth Power the outsides and we get minus X
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cubed the insides and we get plus six x to
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the fourth power and the last time I get minus
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two x squared. And if we want to rewrite
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that in descending powers of X, we get half
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times X equals three x to the fifth power plus
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six x to the fourth power minus X cubed minus
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two X squared. Now let's find the quotient of
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F over G. So we have X cubed plus
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two x squared over three X squared minus one.
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That's about all we can do with that one.
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Now let's talk about the domains. So again,
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the domain of F was all real numbers, and
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the domain of G was all real numbers. And
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when you multiply them, you still get the intersection
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of the domains. So you get all real numbers
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again. Now something changes when we divide, though
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, because we can't divide by zero. We need
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three X squared minus one to not be equal to
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zero. So what would make that equal zero if
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three X squared equaled one. So if X squared
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equals 1/3 So if X was plus or minus the
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square root of 1/3 then we would have a problem
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. We would have zero on the bottom. So
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we're going to say, for the domain is all
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real numbers except X equals plus or minus square 1/3
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. So I'll just say X is not equal to
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plus or minus square root 1/3.