WEBVTT
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Given that π is the function that maps positive real numbers onto the set of real numbers, where π of π₯ is π₯ minus 19.
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And π is the set that maps numbers in the closed interval from negative two to 13 onto the set of real numbers, where π of π₯ is π₯ minus six.
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Evaluate π times π of seven.
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Letβs begin by recalling a little bit of notation.
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π dot π of π₯ or π times π of π₯ simply means we need to multiply the function π of π₯ by the function π of π₯; weβre finding their product.
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So, letβs just find π of π₯ times π of π₯.
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And weβll consider the domain in a moment.
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Weβre told that π of π₯ is equal to π₯ minus 19 and π of π₯ is π₯ minus six.
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So, the product of these two functions is π₯ minus 19 times π₯ minus six.
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Usually, we might look to distribute these parentheses, but weβre not going to worry about that just yet since weβre going to be evaluating this function at a point.
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Before we do, though, we should check the domain of the function.
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The domain of our function π is the set of positive real numbers.
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Whereas the domain, remember, thatβs the input of the function π, is the values of π₯ in the closed interval from negative two to 13.
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We know that the domain of our function must be the intersection of the domains of π and π.
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Thatβs the overlap.
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And so, itβs the set of positive real numbers up to and including 13.
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π₯ can be greater than zero and less than or equal to 13.
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Now, weβre looking to evaluate π times π of seven, in other words, the value of our function when π₯ is equal to seven.
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π₯ is indeed in our domain, so weβre able to evaluate it.
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To do so, we replace π₯ with seven.
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And we find that π times π of seven is seven minus 19 times seven minus six.
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Seven minus 19 is negative 12, and seven minus six is one.
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So, π times π of seven is negative 12 times one, which is negative 12.
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And so, weβve evaluated the product of our functions π and π at a given value of seven.
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Itβs negative 12.