Answer :
Answer:
To find the asymptotes of the graph
=
2
+
1
y=2x+
x
1
, we need to identify where the function behaves in such a way that it approaches infinity or where it is undefined.
Step-by-step explanation:
Vertical Asymptote:
Vertical asymptotes occur where the denominator of a rational function equals zero and the numerator does not simultaneously equal zero (to avoid holes in the graph).
In
=
2
+
1
y=2x+
x
1
:
The vertical asymptote occurs where
=
0
x=0, because the function is undefined at
=
0
x=0 (division by zero).
Horizontal Asymptote:
Horizontal asymptotes describe the behavior of the function as
x approaches positive or negative infinity.
To find the horizontal asymptote:
Compare the degrees of the numerator and the denominator. Since
2
2x (degree 1) grows faster than
1
x
1
(degree -1) as
x approaches infinity or negative infinity, the horizontal asymptote is determined by the leading terms.
Therefore, as
x approaches infinity:
=
2
+
1
≈
2
y=2x+
x
1
≈2x
As
x approaches negative infinity:
=
2
+
1
≈
2
y=2x+
x
1
≈2x
Thus, the horizontal asymptote is
=
2
y=2x.
Summary of Asymptotes:
Vertical asymptote:
=
0
x=0
Horizontal asymptote:
=
2
y=2x
These asymptotes describe the behavior of the function
=
2
+
1
y=2x+
x
1
as
x approaches certain values or infinity.
