1. If x = (x1 , x2 , … , xn) represents a point in a subset A of Rn, and f(x) is exactly one point in Rm, then we say that f is a function from (a domain in) Rn to Rm.
The function f is called a
scalar-valued function if m = 1
vector-valued function if m > 1
function of a single variable if n = 1
function of several variables if n > 1
f(x) = 8 – x is a
scalar-valued function of
a single variable with
domain
{x | x  8} and range
{y | y  0}.
f(x1 , x2 , x3) = x2/x1 +  8 – x32 is a
scalar-valued function of
three
variables with domain
{(x1 , x2 , x3) | x1  0 , – < x2 <  , |x3|  8}
and range
{y | – < y < }.

2. If x = (x1 , x2 , … , xn) represents a point in a subset A of Rn, and f(x) is exactly one point in Rm, then we say that f is a function from (a domain in) Rn to Rm.
The function f is called a
scalar-valued function if m = 1
vector-valued function if m > 1
function of a single variable if n = 1
function of several variables if n > 1
f(x) = (3|x| + 5 , x2) is a
vector-valued function of
a single variable with
domain
{x | – < x < } and range
{(y1 , y2) | y1  5 , y2  0}.
f(x1 , x2) = (x2/x1 ,  8 – x12 , x1x2) is a
vector-valued function of
two
variables with domain
{(x1 , x2) | 0 < |x1|  8 , – < x2 <  } and
range
{(y1 , y2 , y3) | – < y1 <  , 0  y2 < 8 , – < y3 < }.
Definition of the graph of a single-valued function (page 97)
Definition of level curves and level surfaces (page 99)

3. Example 2 (page 98)
z = f(x,y) = x + y + 2 y
The level curves are x + y + 2 = c
c = 0
c = 1
c = –1
c = 1
c = 0
c = –1 x
x + y = – 2
x + y = – 1
x + y = – 3
This is the graph of
a plane.

4. Example 2 (page 98)
z = f(x,y) = x + y + 2
This is a plane.
z = f(x,y)
(0, 0, 2)
(–2, 0, 0) y
(0, –2, 0) x

5. z = f(x,y) = x – y + 2
This is a plane.
z = f(x,y)
(0, 0, 2)
(–2, 0, 0) y
(0, 2, 0) x

6. z = f(x,y) = 3x
This is a plane.
z = f(x,y)
(1, 0, 3) y
(–1, 0, –3) x

7. Example 3 (page 99)
z = f(x,y) = x2 + y2 y
The level curves are x2 + y2 = c
c = 0
c = 1
c = –1
c = 2
c = 2
c = 1
c = 0 x
x2 + y2 = 0
x2 + y2 = 1
x2 + y2 = –1
The level curve is empty.
x2 + y2 = 2

8. Example 3 (page 99)
z = f(x,y) = x2 + y2
Each level curve resulting from letting z be a constant c > 0 is
Each level curve resulting from letting either x or y be a constant c is
This is the graph of
a circle of radius c centered at the origin.
a parabola.
a circular paraboloid (pictured in Figure 2.1.7).
Look at the Conic Sections Handout and the Quadric Surfaces Handout to see how various two-dimensional graphs and three-dimensional graphs can be identified.

9. Example 4 (page 100)
z = f(x,y) = x2 – y2 y
The level curves are x2 – y2 = c
c = 0
c = 1
c = –1 x
x2 – y2 = 0
x2 – y2 = 1
c = 1
x2 – y2 = –1
c = 0
c = –1
This is the graph of
a hyperbolic paraboloid (pictured in Figure ).

10. x2 + 4y2 – z2 = – 4
Each level curve resulting from letting z be a constant c, where |c|>2, is
an ellipse.
Each level curve resulting from letting z be a constant c, where |c|=2, is
a point.
Each level curve resulting from letting z be a constant c, where |c|<2, is
no points at all.
Each level curve resulting from letting x be a constant c is
Each level curve resulting from letting y be a constant c is
a hyperbola.
a hyperbola.
This is the graph of
a hyperboloid of two sheets.

11. x2/9 + y2/16 + z2 = 1
Each level curve resulting from letting z be a constant |c|<1 is
an ellipse.
Each level curve resulting from letting x be a constant |c|<3 is
an ellipse.
Each level curve resulting from letting y be a constant |c|<4 is
an ellipse.
This is the graph of
an ellipsoid.
x2 + y2 – z2 = 4
Each level curve resulting from letting z be a constant c is
a circle.
Each level curve resulting from letting x be a constant c is
a hyperbola.
Each level curve resulting from letting y be a constant c is
a hyperbola.
This is the graph of
a hyperboloid of one sheet.

12. x2 + y2 – z2 = 0
Each level curve resulting from letting z be a constant |c| > 0 is
a circle.
The level curve resulting from letting z be the constant 0 is
the point (0 , 0).
Each level curve resulting from letting x be a constant |c| > 0 is
a hyperbola.
The level curve resulting from letting x be the constant 0 is
two straight perpendicular lines.
Each level curve resulting from letting y be a constant |c| > 0 is
a hyperbola.
The level curve resulting from letting y be the constant 0 is
two straight perpendicular lines.
This is the graph of
right cone.

13. z = – y2
In the yz plane, this graph is
In R3, this graph (extended parallel to the x axis) is
a parabola.
a right parabolic cylinder.
x2 + y2 = 25
In the xy plane, this graph is
In R3, this graph (extended parallel to the z axis) is
a circle (of radius 5 centered at the origin).
a right circular cylinder.
z = x2 + y2 – 4x – 6y + 13 =
(x – 2)2 + (y – 3)2
Each level curve resulting from letting z be a constant c > 0 is
a circle (of radius c entered at (2,3)).
This is the graph of
a shifted circular paraboloid.

14. Describe each graph in R3.
x2 + 3y2 + z2 = 11
x2 + 3y2 + z2 = 0
z2 = 0
x2 + y2 = 9
x2 + y2 = 0
x2 + y2 + z2 = 10
x2 + y2 + z2 + 1 = 0
x2 – y2 = 0
ellipsoid
the single point (0 , 0 , 0)
the xy plane
a circular cylinder of radius 3 centered at the origin
the line which is the z axis
a sphere of radius 10 centered at the origin
no points at all
two planes (x + y = 0 and x – y = 0 )
Level curves help us picture graphs in R3. It is impossible to picture graphs in Rn for n > 3, but level surfaces can be used to give some insight into graphs in R4.

15. Look at each of the following:
Example 5 (page 102)
w = f(x,y,z) = x2 + y2 + z2
Example 6 (page 103)
w = f(x,y,z) = x2 + y2 – z2