Elementary Functions Class Problems

1
.
Given the sets of Natural Numbers, Integers, Rationals and Reals create a diagram that illustrates the containment of these sets.
2
.
Consider Constant Functions, Linear Functions, Affine Functions, Polynomials, Exponential Functions, and Logarithmic Functions. Just as with the number sets above, create a diagram of containment that illustrates intersections of these sets of functions.
3
.
Consider the function P on the Cartesian plane
(
2

)
that is the input being tuples of real numbers that sends the tuple (x,y) to the result x+2y in . Show that this function is linear. That is that for all real numbers
x,y,
x
1
,
x
2
,
y
1
,
y
2
,anda
it satisfies:
3
.
1
.
P(
x
1
+
x
2
,
y
1
+
y
2
)=P(
x
1
,
y
1
)+P(
x
2
,
y
2
)
,
3
.
1
.
1
.
P(
x
1
+
x
2
,
y
1
+
y
2
)=
x
1
+
x
2
+2(
y
1
+
y
2
)=
x
1
+2
y
1
+
x
2
+2
y
2
=P(
x
1
,
y
1
)+P(
x
2
,
y
2
)
3
.
2
.
P(ax,ay)=aP(x,y)
3
.
2
.
1
.
P(ax,ay)=ax+2ay=a(x+2y)=aP(x,y)
4
.
Determine the number of Variables, The degree, and the coefficients of the following polynomials.
4
.
1
.
4
2
x
4
.
1
.
1
.
1 variable, degree 1, [4]
4
.
2
.
x+y
4
.
2
.
1
.
2 variables, degree 1, [1,1]
4
.
3
.
2
2
x
3
y
4
.
3
.
1
.
2 variables, degree 5, [2]
4
.
4
.
2
x
+2xy+
2
y
4
.
4
.
1
.
2 variables, degree 2, [1,2,1]
4
.
5
.
(x-y)(
2
x
+xy+
2
y
)
4
.
5
.
1
.
2 variables, degree 3, [1,1]
4
.
6
.
(x+y)(x-y)+
2
y
4
.
6
.
1
.
1 variable, degree 1, [1]
5
.
Find the roots of the following polynomials:
5
.
1
.
x-1
5
.
1
.
1
.
1
5
.
2
.
(x-1)(x+2)(x-3)
5
.
2
.
1
.
1,-2,3
5
.
3
.
2
x
-4
5
.
3
.
1
.
-2,2
5
.
4
.
(x-3)(
2
x
+3x+9)
5
.
4
.
1
.
3
5
.
5
.
(x+2)(
2
x
-2x+4)
5
.
5
.
1
.
-2
6
.
Determine if
2*
x
3
is an exponential function
6
.
1
.
It is not let
g(x)=2*
x
3
, then
g(0+0)=2*
0+0
3
=2
but
g(0)*g(0)=4
7
.
Determine if 2log(x) is a logarithmic function.
7
.
1
.
It is. Let
f(x)=2log(x)
. Notice that
f(xy)=2log(xy)=2(log(x)+log(y))=2log(x)+2log(y)=f(x)+f(y)
8
.
Consider
log
x
(y)
as a binary function on x and y. Determine whether this function is associative or commutative(symmetric).
8
.
1
.
The question asks whether
log
x
(
log
y
(z))=
log
log
x
(y)
(z)
and if
log
x
(y)=
log
y
(x)
​We can see by examples that this is not the case.
In[]:=
Log[Log[100,10],10]
Out[]=
-
Log[10]
Log[2]
In[]:=
Log[Log[100,Log[10,10]]]
Out[]=
-∞
In[]:=
Log[10,100]
Out[]=
2
In[]:=
Log[100,10]
Out[]=
1
2
​
9
.
Consider
y
x
as a binary function on x and y. Determine whether this function is associative or commutative(symmetric).
9
.
1
.
The question asks whether
z
(
y
x
)
=
(
z
y
)
x
and if
y
x
=
x
y
​
We can see by examples that this is not the case:
In[]:=
(2^3)^4
Out[]=
4096
In[]:=
2^(3^4)
Out[]=
2417851639229258349412352
In[]:=
2^3
Out[]=
8
In[]:=
3^2
Out[]=
9