Step-by-step explanation:
Domain of function
f
(
x
)
=
2
x
x
−
3
is all real numbers except
3
.
Range is all real numbers except
2
.
Explanation:
Graph of
f
(
x
)
=
2
x
x
−
3
is
graph{2x/(x-3) [-20, 20, -10, 10]}
Domain, the set of argument values where the function is defined, is, obviously all real numbers except those where denominator
(
x
−
3
)
is zero, and it happened to be only
x
=
3
. So, for all
x
≠
3
the function is defined and its domain is
x
≠
3
.
One of the ways to determine the range of a function
f
(
x
)
is to consider a domain of an inverse function
f
−
1
(
x
)
. In our case, if
y
=
2
x
x
−
3
then
x
=
3
y
y
−
2
. Therefore, inverse function is
f
−
1
=
3
x
x
−
2
, and its domain is
x
≠
2
. So, the range of the original function is all real numbers except
2
.
Actually, the number
2
is the limit our function is approaching as
x
tends to
+
∞
or
−
∞
.
