Examples Graph of a function
1 examples
1.1 functions of 1 variable
1.2 functions of 2 variables
1.3 normal graph
examples
functions of 1 variable
graph of function f(x, y) = sin(x) · cos(y).
the graph of function.
f
(
x
)
=
{
a
,
if
x
=
1
,
d
,
if
x
=
2
,
c
,
if
x
=
3
,
{\displaystyle f(x)={\begin{cases}a,&{\text{if }}x=1,\\d,&{\text{if }}x=2,\\c,&{\text{if }}x=3,\end{cases}}}
is
{
(
1
,
a
)
,
(
2
,
d
)
,
(
3
,
c
)
}
.
{\displaystyle \{(1,a),(2,d),(3,c)\}.\,}
the graph of cubic polynomial on real line
f
(
x
)
=
x
3
−
9
x
{\displaystyle f(x)=x^{3}-9x\,}
is
{
(
x
,
x
3
−
9
x
)
:
x
is real number
}
.
{\displaystyle \{(x,x^{3}-9x):x{\text{ real number}}\}.\,}
if set plotted on cartesian plane, result curve (see figure).
functions of 2 variables
plot of graph of f(x, y) = −(cos(x) + cos(y)), showing gradient projected on bottom plane.
the graph of trigonometric function
f
(
x
,
y
)
=
sin
(
x
2
)
cos
(
y
2
)
{\displaystyle f(x,y)=\sin(x^{2})\cos(y^{2})\,}
is
{
(
x
,
y
,
sin
(
x
2
)
cos
(
y
2
)
)
:
x
and
y
are real numbers
}
.
{\displaystyle \{(x,y,\sin(x^{2})\cos(y^{2})):x{\text{ , }}y{\text{ real numbers}}\}.}
if set plotted on 3 dimensional cartesian coordinate system, result surface (see figure).
oftentimes helpful show graph, gradient of function , several level curves. level curves can mapped on function surface or can projected on bottom plane. second figure shows such drawing of graph of function:
f
(
x
,
y
)
=
−
(
cos
(
x
2
)
+
cos
(
y
2
)
)
2
{\displaystyle f(x,y)=-(\cos(x^{2})+\cos(y^{2}))^{2}\,}
normal graph
given function f of n variables:
x
1
,
…
,
x
n
{\displaystyle x_{1},\dotsc ,x_{n}}
, normal graph is
(
∇
f
,
−
1
)
{\displaystyle (\nabla f,-1)}
(up multiplication constant). seen considering graph level set of function
g
(
x
,
z
)
=
f
(
x
)
−
z
{\displaystyle g(x,z)=f(x)-z}
, , using
∇
g
{\displaystyle \nabla g}
normal level sets.
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