Answer :
Answer:Conclusion:
Based on the above analysis, the correct answer is:
c. Its domain is
(
6
,
∞
)
and its range is
(
0
,
∞
)
.
c. Its domain is (6,∞) and its range is (0,∞).
Step-by-step explanation:To determine the domain and range of the function
(
)
=
1
−
6
f(x)=
x−6
1
, let's analyze it step by step.
Domain:
The function
−
6
x−6
requires the argument inside the square root to be non-negative. Thus:
−
6
≥
0
x−6≥0
≥
6
x≥6
Additionally, the denominator
−
6
x−6
must be non-zero to avoid division by zero. Therefore:
−
6
≠
0
x−6
=0
−
6
≠
0
x−6
=0
≠
6
x
=6
Combining these conditions, we get:
>
6
x>6
So, the domain of the function is
(
6
,
∞
)
(6,∞).
Range:
Next, we determine the range of the function
(
)
=
1
−
6
f(x)=
x−6
1
. We analyze the behavior of the function as
x approaches the boundaries of the domain.
As
→
6
+
x→6
+
(approaching 6 from the right),
−
6
→
0
+
x−6
→0
+
. Thus:
(
)
=
1
−
6
→
∞
f(x)=
x−6
1
→∞
As
→
∞
x→∞,
−
6
→
∞
x−6
→∞. Thus:
(
)
=
1
−
6
→
0
+
f(x)=
x−6
1
→0
+
The function
(
)
=
1
−
6
f(x)=
x−6
1
is always positive for
>
6
x>6. Therefore, the range of the function is:
(
0
,
∞
)
(0,∞)
