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Determine the domain of the function π of π₯ is equal to π₯ plus four when π₯ is in the closed interval from negative four to eight and π of π₯ is equal to seven π₯ minus 63 when π₯ is in the left-open, right-closed interval from eight to nine.
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In this question, weβre asked to find the domain of a given piecewise function π of π₯.
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And we can start by recalling the domain of any function is the set of input values for that function.
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And for a piecewise-defined function π of π₯, each sub function has its own subdomain.
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And in this case, π of π₯ has two subfunctions: π of π₯ is equal to π₯ plus four whenever π₯ is in the closed interval from negative four to eight and π of π₯ is equal to seven π₯ minus 63 whenever π₯ is in the left-open, right-closed interval from eight to nine.
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Therefore, all of the input values of π₯ are defined by the subdomains.
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The union of the subdomains of a piecewise-defined function is the domain of our function.
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So, the domain of π of π₯ is the union of its two subdomains.
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And thereβs a few different ways of finding the union of these two sets.
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First, to say that π₯ is in the closed interval from negative four to eight is the same as saying π₯ is greater than or equal to negative four and π₯ is less than or equal to eight.
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We can say the same for our other subdomain.
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To say that π₯ is in the left-open, right-closed interval from eight to nine is the same as saying that π₯ is greater than eight or less than or equal to nine.
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To say that π₯ is in the union of these two sets is the same as saying either of these two conditions are true.
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And saying that π₯ is greater than or equal to negative four and less than or equal to eight or π₯ is greater than eight and less than or equal to nine is just the same as saying that π₯ is between negative four and nine.
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And since our inequalities at the end of our expression are not strict, we need to use a closed interval.
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The domain of the function π of π₯ is equal to π₯ plus four when π₯ is in the closed interval from negative four to eight and π of π₯ is equal to seven π₯ minus 63 when π₯ is in the left-open, right-closed interval from eight to nine is the closed interval from negative four to nine.