how do u find the range for this equation
y=x^2+x-2/|x-1|
I went on symbolab, but I dont understand it
thanks
x^2+x = x(x+1)
its minimum is at (-1/2, -1/4)
-2/|x-1| is always negative, and causes an asymptote at x=1
So the range is all reals numbers
it says the range is (-3, infinity )
@oobleck it says the range is (-3, infinity )
To find the range of the equation y = (x^2 + x - 2) / |x - 1|, we need to consider a few things:
1. Determine the domain: The equation contains a term with |x - 1|. For the expression inside the absolute value, x - 1 cannot be equal to zero since it would result in dividing by zero. Therefore, the domain of the equation is all real numbers except x = 1.
2. Analyze the behavior of the equation for x < 1 and x > 1:
For x < 1:
When x < 1, the expression inside the absolute value becomes negative (x - 1 < 0). In this case, the equation can be simplified to y = (x^2 + x - 2) / (1 - x). This is because the absolute value of a negative value is the negative of that value. So, the equation now becomes y = -(x^2 + x - 2) / (x - 1).
Simplifying further, y = -(x + 2), which is a linear equation. The range for x < 1 is all real numbers.
For x > 1:
When x > 1, the expression inside the absolute value becomes positive (x - 1 > 0), and the equation remains as y = (x^2 + x - 2) / (x - 1). This cannot be further simplified.
To analyze the range for x > 1, we can examine the behavior of the equation as x approaches infinity and negative infinity.
As x approaches negative infinity, the equation becomes y = (x^2 + x - 2) / (x - 1) ≈ x, since x^2 and x dominate as x becomes very large. Therefore, as x approaches negative infinity, y also approaches negative infinity.
As x approaches positive infinity, the equation becomes y = (x^2 + x - 2) / (x - 1) ≈ x, because again, x^2 and x dominate as x becomes very large. Consequently, as x approaches positive infinity, y approaches positive infinity.
Combining these observations, we can conclude that the range of the equation is (-∞, ∞), meaning that y can take any real value.
If the explanation is unclear, it is always helpful to consult a math tutor or try other resources that provide step-by-step explanations, such as Khan Academy or a math textbook.