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In the previous lesson we introduced Exponential Functions and their
graphs, and covered an application of Exponential Functions (Compound
Interest). We saw that when interest is compounded 𝑛 times per year for
some number of years, the accumulated value of that investment can be
found by using the formula 𝐴 = 𝑃 (1 +𝑟
𝑛)
𝑛𝑡. Using that formula we can
find the accumulated value of investments when interest is compounded
bi-annually (𝑛 = 2), quarterly (𝑛 = 4), monthly (𝑛 = 12), weekly (𝑛 = 52), etc. But what would happen if the number of compounding
periods per year (𝑛) continued to increase? For instance what if interest
were compounded every day (𝑛 = 365), or every hour (𝑛 = 8760), or
every minute (𝑛 = 525,000), or every second (𝑛 = 31,536, 000)? For a
simple example, imagine that one dollar is invested for one year at a rate
of 100% (𝑃, 𝑡, and 𝑟 = 1); what would that investment accumulate to if
the number of compounding periods continued to increase?
How often is
interest
compounded?
𝒏 𝑨 = 𝑷 (𝟏 +𝒓
𝒏)
𝒏𝒕
monthly 12 𝐴 = 1 (1 +1
12)
12∙1
≈ 2.61303529 …
weekly 52 𝐴 = 1 (1 +1
52)
52∙1
≈ 2.69259695 …
daily 365 𝐴 = 1 (1 +1
365)
365∙1
≈ 2.71456748 …
every hour 8760 𝐴 ≈ 2.71812669161742 …
every minute 525,000 𝐴 ≈ 2.71827924258871 …
every second 31,536,000 𝐴 ≈ 2.71828178130146 …
continuously 𝑛 → ∞ 𝐴 ≈ 2.71828182845902 …
What we see is
that as the number
of compounding
periods per year
increase (𝑛 → ∞),
the accumulated
value 𝐴 continues
to get closer and
closer to the value
2.718281828 … .
This value that we
get closer and
closer to is known
as the natural
number 𝑒, and it is
used in finance (as
we’ll see later
with our second
compound interest
formula), as well
as in other
disciplines such as
statistics and
engineering.
The Natural Number 𝒆:
- a non-terminating constant 2.718281828 …
o to find the value on your calculator press the number 1, then the
2nd function button, then the LN button
- it is NOT a variable like 𝑥 or 𝑦, but rather a number like 𝜋
The way we will use the natural number 𝑒 is either as the base of an
exponential function, which is known as the natural exponential function
𝑓(𝑥) = 𝑒𝑥, or as the base of a logarithmic function, which is known as the
natural logarithmic function 𝑔(𝑥) = ln(𝑥). We will work with the
Natural Exponential Function 𝑓(𝑥) = 𝑒𝑥 in this lesson, and we’ll cover
the Natural Logarithmic Function 𝑔(𝑥) = ln(𝑥) in Lesson 32.
Natural Exponential Functions:
- 𝑓(𝑥) = 𝑒𝑥, where the exponent 𝑥 is a variable and base 𝑒 is the
natural number 2.71828 …
Just like any other exponential function, the domain of the natural
exponential function 𝑓(𝑥) = 𝑒𝑥 is unrestricted (−∞, ∞) and the range is
only positive numbers (0, ∞). And just like any other exponential
function, the natural exponential function 𝑓(𝑥) = 𝑒𝑥 can be transformed,
which in the case of a vertical transformation would alter the range.
Example 1: Given the input/output table for the function 𝑓(𝑥) = 𝑒𝑥, as
well as its graph, find its domain, range, zeros, positive/negative intervals,
increasing/decreasing intervals, and intercepts.
Inputs Outputs
𝑥 𝑓(𝑥) = 𝑒𝑥
𝑥 → −∞ 𝑓(𝑥) → 0
−5 𝑓(−5) = 𝑒−5 =1
𝑒5≈ 0.01
−4 𝑓(−4) = 𝑒−4 =1
𝑒4≈ 0.02
−3 𝑓(−3) = 𝑒−3 =1
𝑒3≈ 0.05
−2 𝑓(−2) = 𝑒−2 =1
𝑒2≈ 0.14
−1 𝑓(−1) = 𝑒−1 =1
𝑒1≈ 0.37
0 𝑓(0) = 𝑒0 = 1
1 𝑓(1) = 𝑒1 ≈ 2.718281 …
2 𝑓(2) = 𝑒2 ≈ 7.389056 …
3 𝑓(3) = 𝑒3 ≈ 20.08553 …
4 𝑓(4) = 𝑒4 ≈ 54.59815 …
5 𝑓(5) = 𝑒5 ≈ 148.4131 …
𝑥 → ∞ 𝑓(𝑥) → ∞
Domain: (−∞, ∞)
Range: (𝟎, ∞)
Zeros: 𝐍𝐎𝐍𝐄
𝑓(𝑥) ≠ 0
Positive intervals: 𝑓(𝑥) > 0 when 𝑥 is (−∞, ∞)
Negative intervals: 𝑓(𝑥) < 0 when 𝑥 is 𝐍𝐎𝐍𝐄
𝐈𝐧𝐜𝐫𝐞𝐚𝐬𝐢𝐧𝐠 𝐢𝐧𝐭𝐞𝐫𝐯𝐚𝐥𝐬: 𝒇(𝒙) 𝐢𝐬 𝐫𝐢𝐬𝐢𝐧𝐠 𝐰𝐡𝐞𝐧 𝒙 𝐢𝐬 (−∞, ∞)
Decreasing intervals: 𝑓(𝑥) is falling when 𝑥 is 𝐍𝐎𝐍𝐄
Intercepts: 𝑥 − intercept: 𝐍𝐎𝐍𝐄 𝑦 − intercept: (𝟎, 𝟏)
𝑓(𝑥) = 𝑒𝑥
Notice that the graph of
𝑓(𝑥) = 𝑒𝑥 is increasing
throughout its domain.
When a function is
always increasing or
always decreasing that
function is one-to-one,
and it will have an
inverse function. All
exponential functions are
one-to-one, therefore all
exponential functions
have an inverse (we’ll
discuss this further in
Lessons 31 & 32).
𝑓(𝑥)
Outputs
Notice that just like the graph of 2𝑥, the graph of 𝑒𝑥 has a horizontal
asymptote at 𝑦 = 0 (the 𝑥-axis). That is because the graphs of both 2𝑥
and 𝑒𝑥 approach the 𝑥-axis, but they never touch it or cross it.
Example 2: Re-write the function 𝑔(𝑥) = 𝑒−𝑥 in terms of
𝑓(𝑥) = 𝑒𝑥. Then find the 𝑦-intercept of 𝑔 and find its graph by
transforming the graph of the original function 𝑓. Enter exact answers
only (no approximations) for the 𝑦-intercept.
𝑓(𝑥) = 𝑒𝑥 𝑓(𝑥)
Outputs
𝑥
Inputs
Inputs Outputs
𝑥 𝑓(𝑥) = 𝑒𝑥
𝑥 → −∞ 𝑓(𝑥) → 0
−2 1
𝑒2≈ 0.14
−1 1
𝑒≈ 0.37
0 1
1 𝑒 ≈ 2.72
2 𝑒2 ≈ 7.39
𝑥 → ∞ 𝑓(𝑥) → ∞
Re − write 𝑔(𝑥) in terms of 𝑓(𝑥):
𝒈(𝒙) =
𝒚 − 𝐢𝐧𝐭𝐞𝐫𝐜𝐞𝐩𝐭:
𝑔(0) =
(𝟎, )
Example 3: Re-write each of the following functions in terms of
𝑓(𝑥) = 𝑒𝑥, then transformation the graph of 𝑓. Also, find the 𝑦-intercept
for each function and enter exact answers only, no approximations.
a. ℎ(𝑥) = 𝑒𝑥 + 2
𝒉(𝒙) =
𝒚 − 𝐢𝐧𝐭𝐞𝐫𝐜𝐞𝐩𝐭:
𝒚 − 𝐢𝐧𝐭𝐞𝐫𝐜𝐞𝐩𝐭:
𝒚 − 𝐢𝐧𝐭𝐞𝐫𝐜𝐞𝐩𝐭:
b. 𝑗(𝑥) = −𝑒𝑥
𝒋(𝒙) =
𝒚 − 𝐢𝐧𝐭𝐞𝐫𝐜𝐞𝐩𝐭:
c. 𝑘(𝑥) = 2𝑒𝑥
𝒌(𝒙) =
𝒚 − 𝐢𝐧𝐭𝐞𝐫𝐜𝐞𝐩𝐭:
𝑓(𝑥) Outputs
𝑥 Inputs
𝑓(𝑥)
Outputs
𝑥 Inputs
𝑥 Inputs
𝑓(𝑥) Outputs
Example 4: Re-write each of the following functions in terms of
𝑓(𝑥) = 𝑒𝑥, then transformation the graph of 𝑓. Also, find the 𝑦-intercept
for each function and ENTER EXACT ANSWERS ONLY,
NO APPROXIMATIONS.
a. 𝑚(𝑥) = 𝑒𝑥−2
𝒎(𝒙) =
𝒚 − 𝐢𝐧𝐭𝐞𝐫𝐜𝐞𝐩𝐭:
𝒋(𝒙) =
b. 𝑛(𝑥) = −𝑒𝑥+1
𝒏(𝒙) =
𝒚 − 𝐢𝐧𝐭𝐞𝐫𝐜𝐞𝐩𝐭:
𝑓(𝑥) Outputs
𝑥 Inputs
𝑓(𝑥)
Outputs
𝑥 Inputs
Answers to Examples:
2. 𝑔(𝑥) = 𝑓(−𝑥), 𝑦 − intercept: (0,1) ;
3a. ℎ(𝑥) = 𝑓(𝑥) + 2, 𝑦 − intercept: (0, 3) ;
3b. 𝑗(𝑥) = −𝑓(𝑥), 𝑦 − intercept: (0, −1) ;
3c. 𝑘(𝑥) = 2𝑓(𝑥), 𝑦 − intercept: (0, 2) ;
4a. 𝑚(𝑥) = 𝑓(𝑥 − 2), 𝑦 − intercept: (0,1
𝑒2) ;
4b. 𝑛(𝑥) = −𝑓(𝑥 + 1), 𝑦 − intercept: (0, −𝑒) ;