# Finding the Domain of a Function

**OK, so suppose we don’t have the graph of a function to look at like in the last section…**

**Can we still find the domain and range?**

Domains: |
Yes (as long as the algebra doesn’tget too hairy… and it won’t for us.) |

Ranges: |
Not really (you usually need thepicture — unless it’s something really basic.) |

**So, we’ll just be doing domains on these — which is really where the action is anyway.**

**Asking for the domain of a function is the same as asking**

**“What are all the possible x guys
that I can stick into this thing?”**

**Sometimes, what you’ll really be looking for is**

**“Is there anything I CAN’T stick in?”**

**Check it out:**

Let’s find the domain of |

**Do you see any x guys that would cause a problem here?**

What about |
? |

**So, x = 3 is a bad guy! Everyone else is OK, though.**

**The domain is all real numbers except 3.**

**What would the interval notation be?**

**When in doubt, graph it on a number line:**

**Do the interval notation in two pieces:**

domain |

**YOUR TURN:**

Find the domain of |

**Sometimes, you can’t find the domain with a quick look.**

**Check it out:**

Let’s find the domain of |

**Hmm… It’s not so obvious!**

**BUT, we are still looking for the same thing:**

The bad x that makesthe denominator 0! |

**How do we find it? Easy!**

**Set the denominator = 0 and solve!**

The domain is |

**TRY IT:**

Find the domain of |
*show work!! |

**How about this one?**

**Square roots — what do we know about square roots?**

** … So, 16 is OK to put in.**

** … So, 0 is OK.**

** … Yuck! But, 3.2 is OK.**

** … Nope! Can’t do it!
**

***We only want real numbers!**

**No negatives are OK!**

**The inside of a radical cannot be negative if we want real answers only (no i guys). So, the inside of a radical has to be 0 or a positive number.**

Set |
and solve it! |

**Now, let’s find the domain of**

So, the domain of |
is |
. |

**TRY IT:**

Find the domain of |
. *Show work!! |

**Here’s a messier one:**

The domain is |
. |

**YOUR TURN:**

Find the domain of |
. *Show work! |