Restrict the domain and codomain of $f$ to make it one-to-one on new domain, onto on new codomain
The function $f(x,y)=3x^2+2y^2-5$
$z=3x^2+2y^2-5$
The domain = $\{(x,y)\in\mathbb{R}^2|x,y\in(-\infty,\infty)\}$
The range = $\{z\in\mathbb{R}|z\in[2,\infty)\}$
One to one means that if $f(x_1,y_1)=f(x_2,y_2)$, then $x_1=x_2,y_1=y_2$
Clearly that is not the case here, take $x=-1,y=-1$, then $f(x,y)=3+2-5=0$, take $x=1,y=1$, then $f(x,y)=3+2-5=0$. Therefore, I would restrict the domain to not include negative values of $x,y$
New domain = $\{(x,y)\in\mathbb{R}^2|x,y\in[0,\infty]\}$
Onto means that every $y\in Y$, where $Y$ is the codomain, has a $x\in X$ that maps to it, where $X$ is the domain.
The range = $\{z\in\mathbb{R}|z\in[2,\infty)\}$
The codomain is just $\{z\in\mathbb{R}\}$
I am having trouble restricting this. I could just set the new codomain to be the range, but is this correct?
Best Answer
If you take $D=\{(x,y)\in\mathbb{R}^2\mid x,y\in[0,+\infty]\}=\{(x,y)\in\mathbb{R}^2\mid x,y\ge0\}$, i.e. the entire first quadrant, as the domain, the function is still not one-to-one. For example, consider $z=10$, which is an element of the range. Then $$3x^2+2y^2-5=10 \iff 3x^2+2y^2=15,$$ which, considering $D$ as the domain, still is a quarter of an ellipse, not a single point. So there are infinitely many points $(x,y)\in D$ satisfying $f(x,y)=10$, and the function is not one-to-one. The same holds for any $z\in\text{your range}$.
So you're right that we have to restrict $x$ because of "$x^2$" in the formula, and the same goes for $y$. But that is not sufficient. This question doesn't have a single answer. Some possible answers are to define the domain as $\{(x,y)\in\mathbb{R}^2\mid x\ge0,y=0\}$ or $\{(x,y)\in\mathbb{R}^2\mid x=0,y\ge0\}$ or $\{(x,y)\in\mathbb{R}^2\mid x\ge0,y=x\}$ or …
I'm also not sure about the codomain that you gave. The codomain could be $\mathbb{R}$, but with this codomain the function is obviously not onto. You're right when you say that to make the function onto the codomain must be precisely the range. But the range of the original function is $[-5,+\infty)$, not $[2,+\infty)$.
Depending on how we interpret the question, you probably are allowed to restrict the codomain even further. But then you'll have to adjust the domain accordingly too. You can't consider $f(x,y)=3x^2+2y^2-5$ as a function from $\{(x,y)\in\mathbb{R}^2\mid x,y\ge0\}$ to $[2,+\infty)$, because in this case some values in the domain, such as $(x,y)=(1,1)$, land outside of the codomain.