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Simplify the function 𝑛 of 𝑥 equals three 𝑥 over 𝑥 plus eight plus six over 𝑥 plus eight, and determine its domain.
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The first thing that we can notice here is that we’re trying to add two fractions together.
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And to add fractions, we must have a common denominator.
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This is true when we’re working with whole numbers.
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It’s also true when we’re working with polynomials or with variables in the denominator.
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In our case, our two fractions already have a common denominator.
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This means that we’re able to add the numerators.
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Three 𝑥 plus six just equals three 𝑥 plus six over 𝑥 plus eight.
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Our question does want us to try and simplify this problem.
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So there’s one other thing we can do.
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We can notice that the three and the six both have a common factor of three.
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If we take out that common factor, we can rewrite three 𝑥 plus six to say three times 𝑥 plus two, all over 𝑥 plus eight.
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Since there’s nothing else that cancels out or can be simplified, this is the simplest form of our function.
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Now we’ll need to determine its domain.
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Remember that our domain is all the possible 𝑥-values.
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We want to ask the question here, “Is there any value for 𝑥 that would not yield a valid result?”
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We should notice that we’re dealing with a fraction and we have a variable in our denominator.
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No fraction can ever have a denominator value of zero because we can’t divide by zero.
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We want to know for what value of 𝑥 with the denominator of this fraction be equal to zero.
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To isolate 𝑥, we’ll subtract eight from both sides, and then we’ll have 𝑥 equal to negative eight.
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What that means is we cannot plug in negative eight into our equation.
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Here’s what would happen.
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We would end up with three times negative six over zero, and that’s impossible.
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We cannot divide by zero.
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What we can say about our domain is this: our domain can be all real numbers with the exception of negative eight, or all reals minus negative eight which would be the case for our function 𝑛 of 𝑥 equals three times 𝑥 plus two over 𝑥 plus eight.