I have this exercise $$z={2x\over y+5}$$, and I am supposed to obtain the domain and range. I understand that the domain is all the pair of $(x,y)$ except $y=-5$ , then the exercise said that the range is $z=R$
I dont understand why z accept all the values, suppose that you want to plot the point $(1,-5)$ you wont be able to plot that point because $y=-5$ its not accepted by the domain so it cant output a value for z (range)
Please help me undestand this or how i get the range for rational functions
Thank you!
$\endgroup$3 Answers
$\begingroup$Yes, if you say $y=-4$ you get $z=2x$ which is a linear function and that one has range the $\mathbb{R}$.
$\endgroup$ 1 $\begingroup$Or, let $x=1$, then $z=\dfrac{2}{y+5}$, as a function of $y$, $z$ takes all values except $z=0$.
But, for $x=0$, $y=1$, $z=\dfrac{2\cdot 0}{1+5}=0$.
So it takes all values.
$\endgroup$ 7 $\begingroup$Domain is the set of all inputs a function can take. As you correctly note, the only thing that makes your $f$ undefined is any input with $y=-5$, hence the domain of $f$ is the entire $\mathbb{R}^2$ with the exception of the line $y=-5$, i.e. the set $$ dom(f) = \{(x,y)|y \ne -5\}. $$
As for the range, it is the set of all points to which $f$ can map something. Note that if $r \in \mathbb{R}$, then $f(r/2, -4)=r$, so the range of $f$ is the entire real line.
UPDATE
For example, to clarify range, suppose you want to check if $\pi$ is in the range of $f$. Note that $$ f(\pi/2, -4) = \frac{2 \cdot (\pi/2)}{(-4)+5} = \pi, $$ so indeed $\pi$ is in the range of $f$. So would be any other real number.
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