SOLUTION: Find the number of values of $x$ for which the expression $\frac{x^2-9}{x^2 + 9} + \frac{1}{x}$ is undefined.

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Question 1203468: Find the number of values of $x$ for which the expression $\frac{x^2-9}{x^2 + 9} + \frac{1}{x}$ is undefined.
Found 4 solutions by josgarithmetic, MathLover1, ikleyn, greenestamps:
Answer by josgarithmetic(39627)   (Show Source): You can put this solution on YOUR website!
To write it like would be better.
Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!

Find the number of values of
,
so,






set denominator equal to zero









the expression is undefined when:




Answer by ikleyn(52864)   (Show Source): You can put this solution on YOUR website!
.

The problem does not describe the domain of the given function.
@MathLover1 solved the problem, assuming that the function is defined over 
complex numbers.


To me, it is strange assumption, since for complex numbers, standard designation "z" 
for a variable is usually used by default.


If "x" is used as a variable, it tells us, by default, that x is a real variable.


For real variable (in the real domain), our function is the sum of two addends,
of which first addend is defined everywhere on the number line,
while the second addend has only one point where it is undefined: x= 0.


So, from this point of view, the answer to the problem's question is 1:
there is one and only one point in the number line, where the given function is undefined.

The post-solution notice for the problem's composer:

        - in such problems,  pointing the domain of a function is  .



Answer by greenestamps(13206)   (Show Source): You can put this solution on YOUR website!


The expression is the sum of two rational expression. To determine the domain, there is no need (i.e., it is a lot of unnecessary work) to combine the two rational expressions into one, as one of the other tutors did. The restrictions on the domain, if there are any, come from the two separate rational expressions.

Restrictions on the domain of a rational expression are determined by values of x that make the denominator equal to 0.

In the first expression, the denominator "x^2+9" is always positive, so there are no restrictions on the domain because of it.

In the second expression, the denominator "x" is obviously equal to 0 only at x=0.

So the only restriction on the domain for the given expression is that x can't be 0.

ANSWER: the expression is undefined for a single value of x
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