Answer:The domain of the function
�
(
�
,
�
)
=
�
�
−
�
−
3
f(x,y)=
xy
−
x
−3 consists of the regions in the first and third quadrants where
�
�
≥
0
xy≥0 and
�
�
−
�
−
3
≥
0
xy
−
x
−3≥0, along with the non-negative part of the
�
x-axis where
�
≥
0
x≥0.
Step-by-step explanation:To sketch the domain of the given function
�
(
�
,
�
)
=
�
�
−
�
−
3
f(x,y)=
xy
−
x
−3, we need to consider the values of
�
x and
�
y for which the function is defined.
The function
�
(
�
,
�
)
f(x,y) involves square roots, so the arguments of the square roots must be non-negative:
For
�
�
xy
to be defined, both
�
x and
�
y must be non-negative or
�
�
≥
0
xy≥0.
For
�
x
to be defined,
�
x must be non-negative or
�
≥
0
x≥0.
Additionally, the expression
�
�
−
�
−
3
xy
−
x
−3 should be defined, which means there are no negative square roots in the expression.
Based on the above conditions, we have the following domain restrictions:
�
�
≥
0
xy≥0 (to ensure
�
�
xy
is defined)
�
≥
0
x≥0 (to ensure
�
x
is defined)
�
�
−
�
−
3
xy
−
x
−3 should not result in taking the square root of a negative number.
Let's consider each condition separately:
�
�
≥
0
xy≥0:
This means that either both
�
x and
�
y are non-negative, or both are non-positive.
The region
�
�
>
0
xy>0 corresponds to the first and third quadrants in the
�
�
xy-plane.
The region
�
�
<
0
xy<0 corresponds to the second and fourth quadrants in the
�
�
xy-plane.
�
≥
0
x≥0:
This means that
�
x must be non-negative, which corresponds to the right half of the
�
x-axis.
�
�
−
�
−
3
xy
−
x
−3 should not result in taking the square root of a negative number:
This condition ensures that
�
�
−
�
−
3
≥
0
xy
−
x
−3≥0 for all points in the domain.
Combining all the conditions, the domain of the function
�
(
�
,
�
)
f(x,y) is the intersection of the regions defined by the conditions above.
To summarize, the domain of the function
�
(
�
,
�
)
=
�
�
−
�
−
3
f(x,y)=
xy
−
x
−3 consists of the regions in the first and third quadrants where
�
�
≥
0
xy≥0 and
�
�
−
�
−
3
≥
0
xy
−
x
−3≥0, along with the non-negative part of the
�
x-axis where
�
≥
0
x≥0.
