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Up until now we have worked with what are known as power functions,
which are functions of the form 𝑓
𝑎
. With a power function, the
base is a variable such as 𝑥 and the exponent 𝑎 is a constant such as 2.
Now we will work with exponential functions 𝑓
𝑥
, where the base
𝑎 is a constant, such as 2 , and the exponent is a variable such as 𝑥. Keep
in mind that while we are inverting the base and the exponent when going
from power functions to exponential functions, those two types of
functions are NOT inverses of each other (more about this later).
Exponential Functions:
𝑥
o the exponent 𝑥 is a variable and base 𝑎 is a constant
o Examples:
𝑥
1
3
𝑥
- there are no restrictions on the exponent 𝑥, but there are restrictions
on the base 𝑎
o 𝑎 cannot be negative
because 𝑥 could be a fraction
1
2
1
2
(or
√
𝑎 ) would be undefined if 𝑎 was negative
o 𝑎 cannot be zero
because 𝑥 could be negative
( 𝑥 = − 1
)
− 1
(or
1
𝑎
) would be undefined if 𝑎 was zero
o 𝑎 cannot be one
because 𝑓
𝑥
would not be a
one-to-one function
𝑥
is always equal 1 , so it is not a 1 − 1 function
An exponential function 𝑓
𝑥
must have a positive base other than 1
𝑎 > 0 and 𝑎 ≠ 1
. Keep in mind that while the base of an exponential
function cannot be negative, an exponential function can be transformed
just like any other function, such as ℎ
𝑥
, which is − 1
𝑥
■
Because the exponent 𝑥 is unrestricted, the domain of an exponential
function is all real numbers (−∞, ∞). To find the range of an exponential
function we can use an input/output table like the one given below.
Example 1: List the domain and range of 𝑓
𝑥
in interval notation.
Inputs Outputs
𝑥
2
− 5
5
− 4
4
− 3
3
− 2
2
− 1
0
1
2
3
4
5
2
The domain of 𝑓
𝑥
is all real numbers
and the range is
only positive numbers
. As shown in the table, anything can go in
for 𝑥, but only positive numbers come out.
Starting at the row
where 𝑥 = 0 and
𝑓
( 0
) = 1 , each of
the following rows
are just multiplied
by 10 :
1 ∙ 10 = 10
10 ∙ 10 = 100
100 ∙ 10 = 1000
So it should make
sense why the top
end of the range is
infinity; because
we are essentially
multiplying by 10
repeatedly when
our inputs are
consecutive
positive integers.
And going the
other direction in
the table, we end
up dividing by 10
repeatedly, which
is why we get
closer and closer
to zero. Keep in
mind that because
we are dividing by
10 repeatedly, we
will never reach
zero, or have
anything less than
zero. That is why
the range is ( 0 , ∞).
Example 3 : Given the exponential function ℎ
𝑥
, list the domain
and range in interval notation.
Keep in mind that the base of this exponential function is 3 , not − 3. So
𝑥
Inputs Outputs
𝑥
2
− 5
5
− 4
4
− 3
3
− 2
2
− 1
1
0
1
2
3
4
5
2
An alternative to making an input/output table for each of these problems
is to break an exponential function that has been transformed into smaller
pieces. For instance, the function 𝑓
𝑥
follows:
𝑥
only produces positive function values, so its range would be
𝑥
(or − 1 ∙ 9
𝑥
) would have nothing but negative function values,
because we would take all the positive function values from 9
𝑥
and negate
each of them, so the range of − 9
𝑥
would be
𝑥
- 8 would take the outputs of − 9
𝑥
and simply add 8 to each
of them
, so its range would be
Again, this can be used as an alternative to making an input/output table
for each function. However if it is easier to identify the range of a
function by seeing the outputs in a table, feel free to stick with making
tables.
Example 4 : Given the exponential function ℎ(𝑥) = (
1
4
𝑥
, list the domain
and range in interval notation.
Example 5 : Given the exponential function ℎ
1
4
𝑥
, list the
domain and range in interval notation.
Example 6 : Given the exponential function 𝑘(𝑥) = − (
1
4
𝑥
− 4 , list the
domain and range in interval notation.
