What is the domain and range of 2x^2-18/x^2+3x-10
for rational functions, y = p(x)/q(x), the domain is all reals except where q(x) = 0
So, here that would be all reals except where (x+5)(x-2) = 0
since there is a horizontal asymptote at y=2, and the graph does not cfross the asymptote, the range would be all reals except y=2
y = 2 + (2-6x)/(x^2+3x-10)
The way you typed it, we are only dividing by x^2 ,so the domain is any real number, except x = 0
If you meant:
(2x^2 - 18)/(x^2 + 3x - 10)
= 2(x-3)(x+3)/( (x+5)(x-2) )
the domain is any real number , x ≠ 2 , -5
To determine the domain and range of the function 2x^2-18/x^2+3x-10, we need to consider two aspects:
1. Domain: The domain represents all the values that x can take in the given function. We want to find the values of x that make the denominator (x^2+3x-10) non-zero because dividing by zero is undefined. So, we need to solve the equation x^2+3x-10 = 0 to find the values that make the denominator equal to zero.
The equation x^2+3x-10 = 0 can be factored as (x-2)(x+5) = 0. From this equation, we find two roots: x = 2 and x = -5. These are the values that make the denominator zero. Therefore, the domain of the function is all real numbers except x = 2 and x = -5.
2. Range: The range represents all the possible values that the function can take. To find the range, we can analyze the behavior of the function as x approaches positive infinity and negative infinity.
As x approaches positive infinity, both the numerator and denominator of the function tend to positive infinity. So, the function approaches positive infinity as x approaches positive infinity.
As x approaches negative infinity, the numerator approaches positive infinity, while the denominator approaches negative infinity. Therefore, the function approaches negative infinity as x approaches negative infinity.
Based on this analysis, we can conclude that the range of the function is all real numbers except for zero.
So, to summarize:
Domain: All real numbers except x = 2 and x = -5
Range: All real numbers except zero.