Find the domain of:
6/(9-4x)
7/(x^2-8x)
(x^2+9x)/(x^3-13x^2+40x)
6/(9-4x)
9-4x =0
9-4x +4x = 0+4x
9= 4x
x =9/4
Domain of x is 9/4
7/(x^2-8x)
x^2 -8x = 0
x(x-8) =0
x =0
x -8 =0
x -8+8 = 0
x =8
domain of x are 0, 8
(x^2+9x)/(x^3-13x^2+40x))
x^3-13x^2+40x =0
x (x^2-13x +40) =0
x(x-5)(x-8) =0
x =0
x = 5
x = 8
Domain of x are 0, 5, and 8
the special values are correct, but the domain is all real numbers except those values.
To find the domain of a rational function, we need to determine the values of x for which the denominator is nonzero. In other words, we want to prevent any division by zero.
1. Domain of 6/(9-4x):
The denominator is 9 - 4x. To find the values of x for which this expression is nonzero, we set the denominator equal to zero and solve for x:
9 - 4x = 0
4x = 9
x = 9/4
So, the domain is all real numbers except x = 9/4.
2. Domain of 7/(x^2-8x):
The denominator is x^2 - 8x. To find the values of x for which this expression is nonzero, we set the denominator equal to zero and solve for x:
x^2 - 8x = 0
x(x - 8) = 0
x = 0 or x - 8 = 0
x = 0 or x = 8
So, the domain is all real numbers except x = 0 and x = 8.
3. Domain of (x^2+9x)/(x^3-13x^2+40x):
The denominator is x^3 - 13x^2 + 40x. To find the values of x for which this expression is nonzero, we set the denominator equal to zero and solve for x:
x^3 - 13x^2 + 40x = 0
x(x^2 - 13x + 40) = 0
The quadratic expression x^2 - 13x + 40 can be factored as (x - 5)(x - 8) = 0. So, we have:
x(x - 5)(x - 8) = 0
From this, we can see that x = 0, x = 5, or x = 8.
Therefore, the domain is all real numbers except x = 0, x = 5, and x = 8.