Find the domain of the function `f(x) = (3x)/(x^2-64)` .Express answer in interval notation.

2 Answers

txmedteach's profile pic

txmedteach | High School Teacher | (Level 3) Associate Educator

Posted on

To find the domain of a function, we are just finding all the values of x for which the function f(x) has an answer! One of the ways of doing this is by finding the set of all x's such that f(x) is undefined, and then we can say that the domain is every number besides those values that we found.

This probem gives an example of this form of finding the domain. In order to have `f(x)` be defined, we need to keep that denominator from reaching zero! Recall, you can NEVER divide by zero.

We can then see that there is nothing else limiting the domain of the function, so we'll have the domain after taking the step of finding which x values will set that denominator to zero.

So, let's solve for when the denominator is zero!

`x^2-64 = 0`

Believe it or not, this is actually a quadratic equation. We can actually factor it like you see here:

`(x+8)(x-8) = 0`

Therefore, either `x+8=0` or `x-8=0` because only one of those needs to be true for the whole expression to be zero! These give us the values of x for which f(x) is undefined:

`x = +-8`

So, our domain, then, is going to be every real number that is not `+-8`.

Now, interval form is not the easiest way to express this, but I'll show it to you after I show you a way that makes more intuitive sense first (D is our domain):

`D={ x|x in RR and x!=+-8}`

There we go! nice and simple, very few characters. You'll see why I'd prefer this over interval notation right now, as we express the domain in interval form:

`D={ x | x in (-oo,-8) or x in (-8,8) or x in (8,oo)}`

And there you go! Your domain and how to solve for it! I hope that helps

micaexel's profile pic

micaexel | High School Teacher | (Level 1) eNoter

Posted on

here is a simple answer, we all know you can NEVER divide by zero so set the x^2 - 64 = 0, and solve....this will tell you what x values you cannot have the answer is

x is all real numbers except for 8 and -8 because those values will make a denominator of zero.

Math Commandment: NEVER DIVIDE BY ZERO