SOLUTION: What is the domain of the radical function f of x is equal to the square root of the quantity 2 times x squared minus 3 times x minus 20 end quantity

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Question 1196095: What is the domain of the radical function f of x is equal to the square root of the quantity 2 times x squared minus 3 times x minus 20 end quantity
Found 3 solutions by MathLover1, greenestamps, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
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f%28x%29=sqrt%282x%5E2-3x-20%29

The domain of a radical function is any x value for which the radicand (the value under the radical sign) is not negative.


will be all values of x that make 2x%5E2-3x-20%3E=+0
2x%5E2-3x-20%3E=+0+ factor
%28x+-+4%29+%282x+%2B+5%29%3E=+0

solutions
%28x+-+4%29++%3E=+0 if +x+%3E=4
or
2x+%2B+5%3E=+0+ if x+%3E=-5%2F2

so domain is
{ x+ element +R : x%3C=-5%2F2 or+x%3E=4 }


Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


It is POSSIBLE that tutor MathLover1 is using a valid method to solve the problem; however, what she APPEARS to be doing is not valid.

The given function is

sqrt%282x%5E2-3x-20%29

The domain of the function is those values of x that make 2x%5E2-3x-20 greater than or equal to 0. To determine those values, factor the quadratic:

%282x%2B5%29%28x-4%29%3E=0

The quadratic is equal to zero when either of those factors is zero.

From there, the other tutor appears to then be solving the problem by finding separately where x-4 is greater than zero and where 2x+5 is greater than zero. That is not a valid method.

One valid method for determining the domain is to use the two zeroes of the function, at x = -2/5 and x = 4, to divide the set of x values into three intervals and determine in which of those intervals the function value is positive. That will show that the function value is positive for x < -2.5 and for x > 4 and negative for -2.5 < x < 4.

Another simple valid method is to recognize that the graph of the quadratic is an upward-opening parabola, so it is negative only between x = -2.5 and x = 4.

Either way, the correct domain is determined: (corrected answer; the endpoints of the intervals are included in the domain): x <= -2.5 or x >= 4

Answer by ikleyn(52788) About Me  (Show Source):
You can put this solution on YOUR website!
.

I came to fix the answer given by @greenestamps.

The correct  ANSWER  is  THIS :   the domain is the set  {x <= -2.5}  OR  {x >= 4}.


In this problem,  endpoints of semi-infinite intervals are included in the domain

(@greenestamps excluded them in his answer).