SOLUTION: Please help me find the domain and write BOTH set-builder and interval notations for f(x)=x^2+x+105 over 5x-45.

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Question 107520This question is from textbook Intermediate Algebra
: Please help me find the domain and write BOTH set-builder and interval notations for f(x)=x^2+x+105 over 5x-45. This question is from textbook Intermediate Algebra

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

f%28x%29=%28x%5E2%2Bx%2B105%29%2F%285x-45%29 Start with the given function


5x-45=0 Set the denominator equal to zero. Remember, dividing by 0 is undefined. So if we find values of x that make the denominator zero, then we must exclude them from the domain.



5x=0%2B45Add 45 to both sides


5x=45 Combine like terms on the right side


x=%2845%29%2F%285%29 Divide both sides by 5 to isolate x



x=9 Divide

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Answer:
So our answer is x=9



Since x=9 makes the denominator equal to zero, this means we must exclude x=9 from our domain

So our domain is:

which in plain English reads: x is the set of all real numbers except x%3C%3E9

So our domain looks like this in interval notation


note: remember, the parenthesis excludes 9 from the domain

If we wanted to graph the domain on a number line, we would get:

Graph of the domain in blue and the excluded value represented by open circle

Notice we have a continuous line until we get to the hole at x=9 (which is represented by the open circle).
This graphically represents our domain in which x can be any number except x cannot equal 9