Multivariate Calculus - Bruce E. Shapiro

52 LECTURE 8. FUNCTIONS OF TWO VARIABLES
Example 8.1 The function y = 3x + 7 has a domain D = R and range R = R.
We can write f : R ↦→ R.
Example 8.2 The function y = x 2 only takes on positive values, regardless of
the value of x. Its domain is still D = R but its range is just the nonnegative
real numbers, which we write as the union of the positive reals and {0}, namely,
R = R ∪ {0}, and we can write f : R ↦→ R ∪ {0}.
Example 8.3 Find the natural domain and range of the function
√
4 − x 2
y =
1 − x .
Solution. We make two observations: (1) the denominator becomes zero (and hence
the function is undefined) when x = 1, and (2) the square root is undefined unless
its argument is non-negative. Hence we require x ≠ 1 and
4 − x 2 ≥ 0 ⇒ 4 ≥ x 2
Of course x 2 can not be negative, so we have
0 ≤ x 2 ≤ 4 ⇒ −2 ≤ x ≤ 2
If we include the earlier restriction x ≠ 1 this becomes
D = {x : −2 ≤ x < 1 or 1 < x ≤ 2} = [−2, 1) ∪ (1, 2]
As the argument x gets closer to 1 the function can take on arbitrarily large (for
x < 1) or large negative (for x > 1) values and hence the range of the function is R,
so that
f : [2, 1) ∪ (1, 2] ↦→ R
Our goal now is to extend our definition of a function to include two variables.
Such functions will have the form
z = f(x, y)
and will associate a point a on the z-axis with points in the xy-plane. Such functions
will represent surfaces, and we will use the following notational observations:
• R 2 represents the set of ordered pairs (x, y), where x, y ∈ R.
• Any plane, such as any of the coordinate planes, is equivalent to R 2 because
there is a one-to-one relationship between the points on a plane and the set of
all real-valued ordered pairs (x, y).
so that we can write
f : (D ⊂ R 2 ) ↦→ (R ⊂ R)
The natural domain of a function of two variables is the set in the xy plane
for which the function definition makes sense. The rules for determining the natural
domain are the same as they are form functions of a single variable:
Revised December 6, 2006. Math 250, Fall 2006