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Find the domain of the real function 𝑓 of 𝑥 is equal to 𝑥 squared minus 19 if 𝑥 is less than or equal to six and 𝑓 of 𝑥 is equal to negative 19 if 𝑥 is greater than six but less than 23.
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So the first thing we need think about is what is the domain of a function.
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Well the domain is the set of all possible independent values.
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So it’s all possible 𝑥 values.
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It’s where the function exists.
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So the first place that we’re told that function exists is when 𝑥 is less than or equal to six.
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Well as we’re told that 𝑥 is any value less than or equal to six, then it means that 𝑥 is any value less.
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So therefore we can say that our lower bound is going to be negative infinity.
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And when we show it on our notation, we have a parenthesis, not a bracket.
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And that’s cause a parenthesis means that it’s not including.
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And our value of negative infinity, it means that it’s not including negative infinity cause negative infinity isn’t an 𝑥 value itself.
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It just tells us that 𝑥 can take up any value in this case less than or equal to six.
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The next part of our function tells us that 𝑥 is greater than six but less than 23.
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Now if we check out both parts of our function, we can see that the breaking point for the piecewise function includes six.
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So therefore, there aren’t gonna be any holes in this function.
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So that means there will be possible values of 𝑥 all the way up to, but not including, 23.
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So again, I use the same notation.
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So we’ve got a parenthesis because it’s not including the 23.
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So therefore, the domain of the real function is as shown here.
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We’ve got negative infinity, 23 inside our parentheses.
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And that’s because there is an outcome for every 𝑥 value between negative infinity and 23, but not including negative infinity and 23.