SOLUTION: Find the domain and range for the function. Use interval notation. f(x)= (x^2 + x - 12)/[(x-3)(x+5)]

Algebra ->  Functions -> SOLUTION: Find the domain and range for the function. Use interval notation. f(x)= (x^2 + x - 12)/[(x-3)(x+5)]      Log On


   

Question 92797: Find the domain and range for the function. Use interval notation.
f(x)= (x^2 + x - 12)/[(x-3)(x+5)]
Answer by mathispowerful(115) About Me  (Show Source):
You can put this solution on YOUR website!

f%28x%29=+%28x%5E2+%2B+x+-+12%29%2F%28%28x-3%29%28x%2B5%29%29

First simplify it:
factor the numerator. we get
%28x%2B4%29%28x-3%29%2F%28%28x-3%29%28x%2B5%29%29,
cancel the common factor x-3
we get %28x%2B4%29%2F%28x%2B5%29
for a rational expression, the denominator can not be 0,
so x can not equal to -5
the domain is any real number except -5.
The range: Let's simplify %28x%2B4%29%2F%28x%2B5%29 further
%28x%2B4%29%2F%28x%2B5%29 = %28x%2B5-1%29%2F%28x%2B5%29
=1-1%2F%28x%2B5%29
let's take a look at the fraction 1%2F%28x%2B5%29
when x is in the neighborhood of -5, the fraction can be positive infinity and
negative infinity, and thus 1-1%2F%28x%2B5%29 can be both positive infinity and
negative infinity.
so the range is from negative infinity to positive infinity.
this conclusion can be verified by the graph of f%28x%29=+%28x%5E2+%2B+x+-+12%29%2F%28%28x-3%29%28x%2B5%29%29 below:
graph%28200%2C200%2C+-5.05%2C-4.95%2C+-1000%2C+1000%2C++%281-1%2F%28x%2B5%29%29%29