the domain of the function f(x)=√(x+1)(x-3)/x-2 is
Answers
Step-by-step explanation:
Assuming you are looking for answers such that x∈R , we can use a simple step by step analysis:
Anything inside a square root (radical) cannot be negative. We are going to keep this in mind.
Through point 1, we can infer that the only part of importance in this question is what is inside the radical sign. So let us continue our analysis to (x−1)(x+2)(x−3)(x−4) .
Since its a purely rational expression, I'm going to use a simple zero/pole analysis. A zero is a value of the independent variable, that makes the expression zero. Similarly, a pole makes the expression undefined (tend to infinity: →∞ ). So our expression yields:
Zeroes: x∈{1,−2}
Poles: x∉{3,4}
4. It is obvious that since we need a real output, our independent variable 'x' can never take values {3, 4}. Also our domain space is now divided into five parts:
−∞<x≤−2
−2≤x≤1
1≤x<3
3<x<4
4<x<∞
5. Now we simply analyse these domain spaces, one-by-one. I'll do one for you (sweat out the rest yourself): 1≤x<3 . Now in this case, the terms (x-1) and (x+2) are positive while (x-3) and (x-4) are (obviously) negative terms. This makes the complete expression of Step 2, positive. Since this result confirms with all of our steps above, we can safely agree that this space belongs in our domain. Similarly, we can see that the interval −2≤x≤1 cannot belong to our domain.
6. The end result is that our domain is the union of all valid spaces. In this particular case, it comes out to be (−∞,−2]∪[1,3)∪(4,∞) .
NOTE: A generalised shortcut to the above approach is the wavy-curve method. Do note that the wavy curve method must be applied very carefully and is valid only for rational expressions.