SOLUTION: Find the domain of the function f(x)= (x+2)/(x^2-9) (write your solution in interval notation)

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Question 103627: Find the domain of the function f(x)= (x+2)/(x^2-9)
(write your solution in interval notation)
Answer by jim_thompson5910(21667) About Me  (Show Source):
You can put this solution on YOUR website!

f%28x%29=%28x%2B2%29%2F%28x%5E2-9%29 Start with the given function


x%5E2-9=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.




%28x-3%29%28x%2B3%29=0 Factor the left side (note: if you need help with factoring, check out this solver)




Now set each factor equal to zero:

x-3=0 or x%2B3=0

x=3 or x=-3 Now solve for x in each case


So our solutions are x=3 or x=-3



Since x=-3 and x=3 make the denominator equal to zero, this means we must exclude x=-3 and x=3 from our domain

So our domain is:

which in plain English reads: x is the set of all real numbers except x%3C%3E-3 or x%3C%3E3

So our domain looks like this in interval notation


note: remember, the parenthesis excludes -3 and 3 from the domain



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

drawing%28500%2C50%2C-10%2C10%2C-10%2C10%2C%0D%0Anumber_line%28+500%2C+-10%2C+10%29%2C%0D%0Ablue%28line%28-2.5%2C-7%2C2.65%2C-7%29%29%2C%0D%0Ablue%28line%28-2.5%2C-6%2C2.65%2C-6%29%29%2C%0D%0Ablue%28line%28-2.5%2C-5%2C2.65%2C-5%29%29%2C%0D%0Ablue%28arrow%283.5%2C-7%2C10%2C-7%29%29%2C%0D%0Ablue%28arrow%283.5%2C-6.5%2C10%2C-6.5%29%29%2C%0D%0Ablue%28arrow%283.5%2C-6%2C10%2C-6%29%29%2C%0D%0Ablue%28arrow%283.5%2C-5.5%2C10%2C-5.5%29%29%2C%0D%0Ablue%28arrow%283.5%2C-5%2C10%2C-5%29%29%2C%0D%0Ablue%28arrow%28-3.5%2C-7%2C-10%2C-7%29%29%2C%0D%0Ablue%28arrow%28-3.5%2C-6.5%2C-10%2C-6.5%29%29%2C%0D%0Ablue%28arrow%28-3.5%2C-6%2C-10%2C-6%29%29%2C%0D%0Ablue%28arrow%28-3.5%2C-5.5%2C-10%2C-5.5%29%29%2C%0D%0Ablue%28arrow%28-3.5%2C-5%2C-10%2C-5%29%29%2C%0D%0A%0D%0Acircle%28-3%2C-5.8%2C0.35%29%2C%0D%0Acircle%28-3%2C-5.8%2C0.4%29%2C%0D%0Acircle%28-3%2C-5.8%2C0.45%29%2C%0D%0A%0D%0A%0D%0Acircle%283%2C-5.8%2C0.35%29%2C%0D%0Acircle%283%2C-5.8%2C0.4%29%2C%0D%0Acircle%283%2C-5.8%2C0.45%29%0D%0A%0D%0A%0D%0A%29 Graph of the domain in blue and the excluded values represented by open circles

Notice we have a continuous line until we get to the holes at x=-3 and x=3 (which is represented by the open circles).
This graphically represents our domain in which x can be any number except x cannot equal -3 or 3