WEBVTT
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Simplify the function π of π₯ equals negative eight divided by π₯ minus six plus π₯ minus six divided by π₯ squared minus six π₯ and determine its domain.
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At first glance, when we look at the function, it appears that all real numbers are valid inputs.
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However, when we look more closely, we can see that some values of π₯ would make the denominators equal to zero, which would give us undefined values.
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So in order to work out which input values generate undefined outputs, we need to find the values of π₯ that give us zero denominators.
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In this case, this involves solving π₯ minus six equals zero and also π₯ squared minus six π₯ equals zero.
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Adding six to both sides of the first equation gives us π₯ equals six.
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Therefore, an input of six would give us an undefined output.
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We can solve the second equation π₯ squared minus six π₯ equals zero by firstly factorising out an π₯.
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This gives us π₯ multiplied by π₯ minus six equal zero.
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Solving this gives us two answers, π₯ equal zero and π₯ minus six equal zero which follows through to give us π₯ equal six.
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This means that the two solutions π₯ equal zero and π₯ equal six give undefined outputs and, therefore, will not be contained within the domain.
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We can go one step further by saying that the domain of π of π₯ is all the real value minus the number zero and six.
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Weβre also required to simplify the function negative eight divided by π₯ minus six plus π₯ minus six divided by π₯ squared minus six π₯.
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In order to do this, our first step is to find a common denominator.
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Factorising out an π₯ on the denominator of the second term leaves us with π₯ minus six divided by π₯ multiplied by π₯ minus six.
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Multiplying the top and bottom of the first term by π₯ gives us a common denominator of π₯ multiplied by π₯ minus six.
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As we now have a common denominator, we can write it as a single fraction: negative eight π₯ plus π₯ minus six divided by π₯ multiplied by π₯ minus six.
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Grouping the like terms on the numerator, negative eight π₯ plus π₯ gives us negative seven π₯.
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And finally factorising out negative one gives us a simplified function of negative seven π₯ plus six divided by π₯ multiplied by π₯ minus six.
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Therefore, the function negative eight divided by π₯ minus six plus π₯ minus six divided by π₯ squared minus six π₯ can be rewritten: negative seven π₯ plus six divided by π₯ multiplied by π₯ minus six with a domain of all real values with the exception of zero and six.