### Home > CALC > Chapter 1 > Lesson 1.2.5 > Problem 1-90

Simplify the following functions.

*Without a calculator*, sketch each graph, showing roots, holes andasymptotes. Then, state the domain in parts (a) and (b) using **interval**notation and the domain in parts (c) and (d) using**set**notation. 1-90 HW eTool (Desmos). Homework Help ✎

Since (*x* − 2) cancelled out, *x* = 2 is the location of a hole.

Now simplify. And notice what was cancelled out.

The *y*-value of the hole can be found by evaluating the simplified function at *x* = 2.

*x*(*x* + 5) did not cancel out, so *x* = 0 and *x* = −5 are vertical asymptotes.

Now find the end behavior. You can long divide or use an approach statement: *x* → − ∞, *y* → 0 and as *x* → ∞, *y* → 0. There is a horizontal asymptote at *y* = 0.

Last but not least, investigate what is happening in the middle, by exploring the holes and vertical asymptotes. Use the simplified equation:

*x* → −5^{−}, *y* → −∞

*x* → 0^{+}, *y* → +∞

*x* → −5^{+}, *y* → +∞

*x* → 0^{−}, *y* → −∞

Use the eTool below to view the steps for graphing the first equation.

Click the link at right for the full version of the eTool: Calc 1-90 HW eTool