How do you find the domain and range of #y = (x - 3)/( x^2 - 4)#?
2 Answers
Domain:
or
Range:
Explanation:
Domain is easy. This function is defined everywhere except those values of argument
or
Solutions are
At these points the function has vertical asymptotes.
To address the range, let's first transform the function as follows:
Next step is to graph this function as a sum of two graphs
The resulting graph would look like
graph{1/(x+2)-1/(x^2-4) [-10, 10, -5, 5]}
It's easy to see from this graph that the only segment not covered by values of this function is the one between the maximum of the right part of a graph and minimum of the middle part.
To find these values, let's use the calculus to find the arguments where our function reaches its local maximum and minimum values, that is those values of
Taking the derivative of this function results in'
Now we have solve the equation
or
or
or
In this case we have to find where the numerator equals to zero, that to solve
or
or
Solutions of this quadratic equation are
We have to use the larger value
We have to use the smaller value
Therefore, the range of our function is
Solve for
Explanation:
Here's an alternative method to try to find the range using elementary methods.
Multiply both sides by
Subtract
If
Otherwise, use the quadratic formula to get:
This will have Real solutions when the discriminant is non-negative, that is when:
This quadratic in
Hence