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Given this function:

$$f(x,y)=\sqrt{y-1-x}+\ln{(x-y^2+4y-3)}$$

Characterize the domain.

I isolated $y$ for the expressions under the square root and inside the natural logarithm. I got the following:

enter image description here

The domain is the space where the two graphs overlap, right? Around $y=2$. How do I describe this accurately? My take is that the domain is limited but continuous.

  • 1
    you have two conditions $$y-1-x\geq 0$$ and $$x-y^2+4y-3>0$$ – 2017-02-25
  • 0
    So I should solve the inequalities for $y$ to get my domain? – 2017-02-25
  • 0
    this is possible, or you make a 2D plot – 2017-02-25
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    post your results and we can compare ours – 2017-02-25

1 Answers 1

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The easiest way to describe the domain of $f$ is by stating a $y$-dependent interval for the allowed values of $x$: $$ y^2-4y+3 < x \leq y-1 $$ You could if you wish add the condition that $1\leq y \leq 4$ but that would be redundant, since it is always satisfied whenever $y^2-4y+3 < x \leq y-1$ can be satisfied for any $x$.