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Determine the domain of the function defined by the given equation.$r(x)=\frac{2 x+1}{x^{3}-9 x}$

$$x \neq-3,0,3$$

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

Oregon State University

McMaster University

Harvey Mudd College

Lectures

01:43

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).

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Find the domain of the giv…

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Find the domain of the fun…

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Find the domain of each ra…

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Find the domain of the exp…

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for this problem. We've been given a function R of X, and we want to find its domain well, to answer that, let's take a step back and review the definition of domain. The domain of a function are all of the inputs that we're allowed to put into our function. Another way to think about this means the same thing, just some different words. They're the X values that are valid to put into our function. Now, when we're looking at these numbers that we can put in, we want to know all the numbers that are possible. It's actually often easier to answer the reverse question. Which numbers are not possible. If we can figure out which numbers we have toe exclude, then everything else is included in the domain. Now there are two important things to check when we're looking for invalid domain numbers. The first is if you have a square root. Since these air real numbers were dealing with the value of what's under the square, root must be positive or zero can't be negative, so we'd have to check and exclude anything that made that value under the radical negative. We also need to Look, if we have fractions, the denominators of fractions cannot be zero, cause I can't divide by zero. Well, in this case, if you look at our function are I don't have any square roots. But I do have a fraction now. The numerator could be anything at once. We don't have to worry about that. It's just the denominator. So let's look at that denominator X cubed minus nine x. And I want to know what values make that equal to zero will exclude those from our domain. Well, this is fact herbal. If I take out an X, I get X squared minus nine equals zero. And in parentheses there. That's the difference of two squares. So I can factor that into X plus three and X minus three. So three factors that gives me three zeros from the first factor X equals zero. The second factor gives me X equal negative three. And the third one gives me X equals positive three. So I need to exclude those three numbers from my domain. Any one of those three means I'm dividing by zero. Every other real number, though, is perfectly valid. These air, the Onley exceptions. So my domain is all real numbers except for the three re found, except for X equals zero negative three or positive three.

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