What is the domain of the real-valued function f(x) = (2x - 7)/sqrt(x^2 - 5x)?
+9466 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, ∞)
