Question

What is the domain of the function?

Answer 1

I got a domain and range of:
(
−
∞
,
5
)
∪
(
5
,
∞
)
, or
x
≠
5
(
−
∞
,
1
)
∪
(
1
,
∞
)
, or
y
≠
1
The function is undefined for
x
values when the denominator,
x
−
5
, is
0
; it's undefined to divide by
0
. Therefore, when
x
=
5
,
f
(
x
)
is undefined.
f
(
5
)
=
5
+
7
5
−
5
=
12
0
Since the domain is based on the allowed values of
x
, the domain is:
(
−
∞
,
5
)
∪
(
5
,
∞
)
Based on the domain, we would find the range by solving for
x
in terms of
f
(
x
)
, which we will write as
y
=
f
(
x
)
.
y
=
x
+
7
x
−
5
y
(
x
−
5
)
=
x
+
7
x
y
−
5
y
=
x
+
7
x
−
x
y
=
−
5
y
−
7
x
(
1
−
y
)
=
−
5
y
−
7
x
=
−
5
y
−
7
1
−
y
x
=
5
y
+
7
y
−
1
This means when
y
=
1
, the function is undefined as well. So, the range is:
(
−
∞
,
1
)
∪
(
1
,
∞
)
You can see that this is the case in the graph itself:
graph{(x + 7)/(x - 5) [-73.3, 74.9, -37.07, 36.97]}
What you should notice is the horizontal asymptote at
y
=
1
, and the vertical asymptote at
x
=
5
.
Because the function is trying to reach an undefined value at those points (
x
≠
5
,
y
≠
1
), you get these "walls" that cannot be crossed, only ascended or descended from either side.