# Find range of exponent

I have the following function I need to find the range for and I'm not sure if I'm on the right direction.

$$f(x,y) = e^{-x^2-(y-1)^2}$$

$$x$$ & $$y$$ are real-numbers.

I'm thinking that the range is "all real values for $$y$$ that are $$> 0$$."

Is this right?

• I think you need to find the range of f(x,y), not y. You need to think about what all values f(x,y) can take for any real x, y – Neo Oct 11 at 9:33

Since $$-x^2-(y-1)^2\leq0$$ and $$g(x)=e^x$$ increases, we obtain: $$0
Note that: $$f(x,y) = e^{-x^2-(y-1)^2}=e^{-(x^2+(y-1)^2)},(x,y)\in \mathbb R^2 \iff \\ f(t)=e^{-t}, t\ge 0.$$ Since $$f(0)=1$$; $$f'(t)=-e^{-t}<0, t>0$$ and $$\lim_\limits{t\to +\infty}f(t)=0$$, we get: $$0