Which of the following exponential functions goes through the points (1, 6) and (2, 12)? f(x) = 3(2)x f(x) = 2(3)x f(x) = 3(2)−x f(x) = 2(3)−x

y
=
3
⋅
2
x
Explanation:
All exponential functions can be written in the form
y
=
a
b
x
. We just need to solve for
a
and
b
.
Since the function goes through the points
(
1
,
6
)
and
(
2
,
12
)
,
x
=
1
⇒
y
=
6
and
x
=
2
⇒
y
=
12

(
1
)
.
Since
x
=
1
⇒
y
=
6
, we just need to input these values into the equation
y
=
a
b
x
to get
6
=
a
b
.
Since
x
=
2
⇒
y
=
12
, we just need to input these values into the equation
y
=
a
b
x
to get
12
=
a
b
2

(
2
)
.
Since neither
a
nor
b
can be zero, we can safely divide equation
(
2
)
by equation
(
1
)
to get
12
6
=
a
b
2
a
b
, or
2
=
b
. Since
6
=
a
b
, it must be the case that
a
=
3
.
Our final exponential function is
y
=
3
⋅
2
x
.
We can rewrite this function in several other forms, such as
y
=
3
⋅
2
x
=
2
ln
(
3
)
⋅
2
x
=
2
x
+
ln
(
3
)
or
y
=
3
⋅
2
x
=
3
⋅
e
x
ln
(
2
)
=
3
⋅
10
x
log
10
(
2
)
.