Domain and range stuff

Jan 2020
2
0
Canada
Hi! Doing a course in pre-cal and I messed up on a question. Could someone tell me why f(x) = √x + 4/x - 4 has a domain of [-4, 4] U [4, infinity)

Thanks!
 

romsek

Math Team
Sep 2015
2,766
1,545
USA
do you mean $\sqrt{\dfrac{x+4}{x-4}}$ ?

remember that $\sqrt{x}$ is only defined for $x \geq 0$
 
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Jan 2020
2
0
Canada
do you mean $\sqrt{\dfrac{x+4}{x-4}}$ ?

remember that $\sqrt{x}$ is only defined for $x \geq 0$
yes! sorry, I'm new and haven't really figured out the codes when typing in math equations
 

romsek

Math Team
Sep 2015
2,766
1,545
USA
just as a double check, we want $y = \dfrac{x+4}{x-4} \geq 0$

There are 2 suspect points, where things go to 0, $x=-4, ~x=4$

So we split the real line up into $(-\infty, -4) \cup [-4] \cup (-4, 4) \cup [4] \cup (4, \infty)$

we can make a table of things

$\begin{matrix}
\text{Interval}&\text{Quotient}&\text{Result}\\
(-\infty, -4) &\dfrac N N &P\\
[-4] &\dfrac 0 N&0\\
(-4,4)&\dfrac P N &N\\
[4] &\dfrac P 0 &\text{not defined}\\
(4, \infty)&\dfrac P P &P
\end{matrix}$

and we see that the legit domain is

$(-\infty, -4] \cup (4, \infty)$

Which incidentally doesn't match your original post.
 
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