Question

How do you sketch the curve for `y= (x^2+1)/(x^2-4)`?

Answer 1

Find the critical numbers, local maximum and determine the behaviors as `+-oo`.

Explanation:

Step 1. Determine the domain and range of `f(x)=(x^2+1)/(x^2-4)`.

Domain is all values of `x` except where
`x^2-4=0`
`x^2=4`
`x=+-2`

Range is `(-oo,oo)`

Step 2. Find the critical numbers of `f(x)` in `(-2,2)`.
Apply the Quotient Rule to differentiate the function.
`f'(x)=((x^2-4)(2x)-(x^2+1)(2x))/(x^2-4)^2`
`f'(x)=(2x)/(x^2-4)-(2x(x^2+1))/(x^2-4)^2`

Critical numbers:
`x=+-2`, where `f'(x)` does not exist, and `x=0`, where `f'(x)=0`.

Step 3. Calculate `f(x)` with critical numbers.
`f(-2)` does not exist. The limit from the left is `+oo` and the limit from the right is `-oo`.
`f(2)` does not exist. The limit from the left is `-oo` and the limit from the right is `+oo`.
`f(0) = -1/4` is a local maximum.
For `x<-2`, `f(x)>0`. For `x>2`, `f(x)>0`.

Step 4. Sketch the graph with these attributes.

graph{(x^2+1)/(x^2-4) [-20, 20, -10, 10]}