# 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 theÂ domain of a functionÂ is the same as asking**

**“What are all the possibleÂ xÂ 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 anyÂ xÂ guysÂ that would cause a problem here?**

What about |
? |

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

**The domain is all real numbers exceptÂ 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 theÂ domainÂ of |

**Hmm…Â It’s not so obvious!**

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

The badÂ xÂ that makesthe denominatorÂ 0! |

**How do we find it?Â Easy!**

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

TheÂ domainÂ is |

**TRY IT:**

Find the domain of |
*show work!! |

**How about this one?**

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

**Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â …Â So,Â 16Â is OK to put in.**

**Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â …Â So,Â 0Â is OK.**

**Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â …Â Yuck!Â But,Â 3.2Â is OK.**

**Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â …Â Nope!Â Can’t do it!
**Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â

***We only want real numbers!**

**No negatives are OK!**

**TheÂ insideÂ of aÂ radicalÂ cannot be negative if we wantÂ real answersÂ only (noÂ iÂ guys).Â So, theÂ insideÂ of aÂ radicalÂ has to beÂ 0 or a positive number.**

Set |
Â | Â and solve it! |

**Now, let’s find the domain of**

So, the domain of |
is |
. |

**TRY IT:**

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

**Here’s a messier one:**

The domain is |
. |

**YOUR TURN:**

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