Answer
(a) $f\left( {x,y} \right) = |x| + |y|$ ${\ }$ matches Figure (D)
(b) $f\left( {x,y} \right) = \cos \left( {x - y} \right)$ ${\ }$ matches Figure (C)
(c) $f\left( {x,y} \right) = \frac{{ - 1}}{{1 + 9{x^2} + {y^2}}}$ ${\ }$ matches Figure (E)
(d) $f\left( {x,y} \right) = \cos \left( {{y^2}} \right){{\rm{e}}^{ - 0.1\left( {{x^2} + {y^2}} \right)}}$ ${\ }$ matches Figure (B)
(e) $f\left( {x,y} \right) = \frac{{ - 1}}{{1 + 9{x^2} + 9{y^2}}}$ ${\ }$ matches Figure (A)
(f) $f\left( {x,y} \right) = \cos \left( {{x^2} + {y^2}} \right){{\rm{e}}^{ - 0.1\left( {{x^2} + {y^2}} \right)}}$ ${\ }$ matches Figure (F)
Work Step by Step
(a) We have $f\left( {x,y} \right) = |x| + |y|$. By setting $y=a$ we fix the $y$-coordinate and obtain the vertical trace curve $f\left( {x,a} \right) = z = |x| + |a|$ that lies in the plane parallel to the $xz$-plane. Similarly, by setting $x=a$ we fix the $x$-coordinate and obtain the vertical trace curve $f\left( {a,y} \right) = z = |a| + |y|$ in the plane parallel to the $yz$-plane. So, it matches Figure (D).
(b) We have $f\left( {x,y} \right) = \cos \left( {x - y} \right)$. It matches Figure (C).
(c) We have $f\left( {x,y} \right) = \frac{{ - 1}}{{1 + 9{x^2} + {y^2}}}$. It matches Figure (E).
(d) We have $f\left( {x,y} \right) = \cos \left( {{y^2}} \right){{\rm{e}}^{ - 0.1\left( {{x^2} + {y^2}} \right)}}$. It matches Figure (B).
(e) We have $f\left( {x,y} \right) = \frac{{ - 1}}{{1 + 9{x^2} + 9{y^2}}}$. It matches Figure (A).
(f) We have $f\left( {x,y} \right) = \cos \left( {{x^2} + {y^2}} \right){{\rm{e}}^{ - 0.1\left( {{x^2} + {y^2}} \right)}}$. It matches Figure (F).
