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Find the domain of the function π of π₯ equals π₯ plus four over π₯ minus eight squared.
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What is the domain of our function?
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Itβs the set of values of π₯ on which π of π₯ is defined.
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We could ask ourselves, is two in the domain?
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Thatβs the same as asking, is π of two defined?
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Well, letβs find out.
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We use our definition of π of π₯.
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Replacing π₯ by two, we get that π of two is equal to two plus four over two minus eight squared.
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In the numerator, two plus four is six.
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And in the denominator, two minus eight is negative six.
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And that squared is 36.
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So we get six over 36 which is a sixth.
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So clearly, π of two is defined.
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And so two must be in the domain.
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Letβs try another number.
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Is eight in the domain?
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Thatβs the same as asking if π of eight is defined.
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Substituting eight in, we get that π of eight is eight plus four over eight minus eight squared.
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In the numerator, eight plus four is 12.
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And in the denominator, eight minus eight is zero.
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And zero squared is also zero.
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And so we get 12 over zero.
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12 over zero is not defined.
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You canβt divide a number by zero.
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So π of eight is not defined.
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And eight is therefore, not in the domain of our function.
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Okay, so we canβt go through this process for every real number.
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Weβre going to have to be slightly cleverer than that.
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There was no problem with two.
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Two was in the domain.
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What was the problem was eight?
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Well, the problem was that we ended up with a denominator of zero.
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And we canβt divide by zero.
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It turns out that for a rational function, that is a function which is written in the form of a fraction where both the numerator and the denominator are polynomials, this is the only thing that can go wrong.
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The domain is all real numbers except those values which make the denominator zero.
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The denominator of our function is π₯ minus eight squared.
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So our domain is the set of real numbers minus the set of values of π₯ which make π₯ minus eight squared equal to zero.
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Which values of π₯ make π₯ minus eight squared equal to zero?
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Well, only π₯ equals eight.
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Our domain is therefore, all real numbers apart from eight, which is written in set notation like this.
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For any value of π₯ apart from eight, which makes the denominator zero, the rational function π of π₯ is defined.