I'm stuck on this problem please help me
find the range of this equation
y = (x^2 - 4)/(x^2 - x - 12)
I know that it is [1/3,-infintiy]U[?,infinity]
yes I'm having trouble with that ?
At first I was like ok there should be a horizontal asymptote at y=1 there is in calculus deffinition the image as x approaches infinity from both sides right? or is just infinity... but ya it appraoches one from both sides but it corsses one at x=-8 check the graph so how do I find how low it goes at this section also I can't use a calculator well I can but am suppose to show all work with out using one i.e. have one to use not suppose to use it thanks
also unless I'm doing something wrong when x=100 y is not one you told me it did I must be doing something wrong
To find the range of the equation y = (x^2 - 4)/(x^2 - x - 12), we need to determine the values that y can take on as x varies.
To start, let's analyze the denominator, (x^2 - x - 12). We want to find the values of x that make the denominator equal to zero, as those values will create undefined points in the function. We can factor the denominator to get: (x + 3)(x - 4) = 0. This means that x = -3 and x = 4 are the values that make the denominator equal to zero.
Now, let's look at the numerator, (x^2 - 4). We can factor this expression as well to get: (x + 2)(x - 2). This shows us that x = -2 and x = 2 are the values that make the numerator equal to zero.
So, we have determined that x = -3, x = 4, x = -2, and x = 2 are the values of x that create undefined points in the function.
Next, we can consider the behavior of the function as x approaches positive and negative infinity. As x approaches positive infinity, both the numerator and denominator of the equation tend towards positive infinity. Therefore, the limit of the function as x approaches positive infinity is positive infinity.
Similarly, as x approaches negative infinity, both the numerator and denominator of the equation tend towards positive infinity. Therefore, the limit of the function as x approaches negative infinity is also positive infinity.
This means that the range of the function includes positive infinity.
To find the lower bound of the range, we can consider the behavior of the function in the vicinity of x = -8, where it crosses the horizontal asymptote y = 1. By plugging in x = -8 into the equation, we get y = (-8^2 - 4)/(-8^2 - (-8) - 12) = 64/84. Simplifying this gives us y = 16/21.
Therefore, the lower bound of the range is 16/21.
Combining the information, the range of the equation y = (x^2 - 4)/(x^2 - x - 12) is [16/21, infinity) since y approaches infinity as x approaches infinity.