Functions of Several Variables — Question 8

Question 8 – Solution

Strategy Use $2|xy|\le x^2+y^2$ and then restrict to the lines $y=\pm x$.

Step 1: Bound Let $r^2=x^2+y^2$. Then $$|f|\le\frac{r^2/2}{1+r^2}<\frac 12.$$ Thus the range is contained in $(-1/2,1/2)$.

Step 2: Realize values On $y=x$, $$f(x,x)=\frac{x^2}{1+2x^2},$$ which increases from $0$ toward $1/2$. Solving $c=x^2/(1+2x^2)$ gives $x^2=c/(1-2c)$ for every $0<c<1/2$. The line $y=-x$ similarly realizes every negative value, and $(0,0)$ realizes $0$.

Step 3: Conclusion Therefore $$\boxed{\operatorname{range}(f)=(-1/2,1/2)}.$$ The denominator contains the extra positive $1$, making equality in the bound impossible for finite inputs; the endpoints occur only as limits.