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Consider

$$ f(x) = \sqrt{2-x} - \frac{1}{9-x^2} $$

Now for the radical to be definable,

$$ 2-x \ge 0 $$

And for the fraction to be definable,

$$ 9-x^2> 0 $$

SO , the number line of solution set looks like:

enter image description here Where red line is for radical's definability & orange for fraction's.

Now,

The result from above stuff is

$$ (- \infty , -3) $$

EDIT: I got that the second inequality's solution set was wrong.

Its

$$ (-3,3)$$

So final solution plot:

Green: 1st inequation and red for 2nd. Blue: final solution.

Which is $ \color{red}{wrong} $

Why?

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  • $\begingroup$ A fraction is defined when the denominator is not equal to zero. $\endgroup$ – N. F. Taussig May 23 '18 at 10:28
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The fraction is defined if and only if $9-x^2\ne 0\Leftrightarrow x\ne\pm3.$

However, you have done the square root part correctly: $\sqrt{2-x}$ is defined if and only if $x\le 2$.

Because of this, we can conclude that $f(x)$ is defined if and only if $x\le 2$ and $x\ne -3$.

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We have to satisfy

  • $2-x\ge 0 \implies x\le 2$

and

  • $9-x^2\neq0\implies x\neq \pm3$

thus the set of values of $x$ which satisfy both inequalities is

  • $x\le2 \land x\neq-3$
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  • $\begingroup$ +1 for nice presentation! However, for something this basic you should probably expand "thus" to something like "thus the set of values of $x$ that satisfy both inequalities is". I've certainly had students sometimes correctly obtain multiple inequalities in domain questions such as this, but then answer with the union (not intersection) of the sets involved. $\endgroup$ – Dave L. Renfro May 23 '18 at 7:14
  • $\begingroup$ @DaveL.Renfro Thanks for your kind suggestion, I’ll add some more details. $\endgroup$ – user May 23 '18 at 7:17
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    $\begingroup$ The expression $\frac{1}{9 - x^2}$ is defined unless $|x| = 3$. Why are you writing $9 - x^2 > 0$? $\endgroup$ – N. F. Taussig May 23 '18 at 10:26
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In one line:

$ x \le 2$ AND $(x \not = -3$ OR $x \not =3)$.

Hence $x \le 2$ AND $x \not = -3.$

$D= (-\infty, -3)\cup (-3,2]$.

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