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# domain

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What is the domain of the real-valued function f(x) = (2x - 7)/sqrt(x^2 - 5x)?

Mar 6, 2021

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Values of  x  that make the denominator 0 must be excluded from the domain.

Values of  x  that make the expression inside the square root negative must be excluded from the domain.

So.... any values of  x  that make  x2 - 5x  less than or equal to zero must be excluded from the domain.

x2 - 5x  ≤  0

To find which values of  x  make this negative, first let's find which values make it equal zero.

x2 - 5x  =  0

x(x - 5)  =  0

x  =  0     or     x  =  5

The graph of   y = x2 - 5x  is a parabola that intersects the  x-axis when  x = 0  and  x = 5.

So to find which values of  x  make the expression negative, we can test a point in each of the following intervals:

(-∞, 0)  or  (0, 5)  or  (5, ∞)

(-1)2 - 5(-1)   =   1 + 5   =   6   >   0    So the values of  x  in (-∞, 0)  make the expression positive

(1)2 - 5(1)   =   1 - 5   =   -4   <   0   So the values of  x  in  (0, 5)  make the expression negative

(6)2 - 5(6)   =   36 - 30   =   6   >   0  So the values of  x  in  (5, ∞)  make the expression positive

And so the values of  x  that fall in the range [0, 5]  must be excluded from the domain.

So the domain is:

(-∞, 0)  U  (5, ∞)

Mar 6, 2021