The document discusses exponential functions and their applications to modeling real-world situations involving population growth, radioactive decay, and compound interest. Exponential functions have the variable in the exponent and have properties like being one-to-one that allow equations with them to be solved. Examples are worked through to demonstrate how to set up and solve exponential models for situations involving doubling populations, radioactive half-life, and compound interest calculations.

1. General Mathematics
CHAPTER 4:
Exponential Function
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2. Why is it important to understand function?
It is important for us to learn exponential function because it help
us to model many real life situations.
Exponential Models and Population Growth
Suppose a quantity 𝑦 doubles every 𝑇 units of time. If 𝑦0 is t
he initial amount, then the quantity 𝑦 after 𝑡 units of time is
given by 𝑦 = 𝑦0(2)
𝑡
𝑇

3. 1. Population
Let t = time in days. At t = 0, there were initially 20 bacteria. Suppose that the bacteria
doubles every 100 hours. Give an exponential model for bacteria as a function of t.
Answer: Initially,
at t = 0, Number of bacteria = 20
at t = 100 Number of bacteria = 20(2)
at t = 200 Number of bacteria = 20(2)2
at t = 300 Number of bacteria = 20(2)3
at t = 400 Number of bacteria = 20(2)4
Therefore, the exponential model for this situation is 𝑦 = 20(2)𝑡/100
Why is it important to understand function?

4. 2. Exponential Decay
Definition.
The HALF-LIFE of a radioactive substance is the time it takes for half of the substance to decay.
Suppose that the half-life of a certain radioactive substance is 10 days and there are 10g
initially, determine the amount of substance after 30 days.
Answer: Initially,
at t = 0 Amount of substance = 10g
at t = 10 Amount of substance = 5g
at t = 20 Amount of substance = 2.5g
at t = 30 Amount of substance = 1.25g
Therefore, the exponential model for this situation is 𝑦 = 10(
1
2
)𝑡/10

5. 3. Compound Interest
Mrs. Dela Cruz invested P100, 000 in a company that offers 6% interest compounded annually.
How much will this investment be worth at the end of each year for the next five years?
Answer: Initially,
at t = 0 Investment = P100, 000
at t = 1 Investment = P100, 000 (1.06)
at t = 2 Investment = P100, 000 (1.06)2
at t = 3 Investment = P100, 000 (1.06)3
at t = 4 Investment = P100, 000 (1.06)4
at t = 5 Investment = P100, 000 (1.06)5
Therefore, the exponential model for this situation is y = 100, 000(1.06)t
Compound Interest
If a principal P is investe
d at an annual rate of 𝑟,
compounded annually,
then the amount after 𝑡
years is given by
𝐴 = 𝑃(1 + 𝑟)𝑡

6. Exponential Function
• An exponential function is an equation which the variable
appears in an exponent.
• An exponential function with base b is a function of the form
𝑓 𝑥 = 𝑏𝑥 𝑜𝑟 𝑦 = 𝑏𝑥, where 𝑏 > 0, 𝑏 ≠ 1.

7. Graph 𝒇 𝒙 = 𝟐𝒙
The graph starts off slow but then increases rapidly

8. Graph 𝒇 𝒙 = (
𝟏
𝟐
)𝒙
The graph starts of very high but then decreases very rapidly

9. EXAMPLE
𝒇 𝒙 = 𝒃𝒙
𝐰𝐡𝐞𝐫𝐞 𝐛 > 𝟎 𝐚𝐧𝐝 𝐛 ≠ 𝟏 𝒂𝒏𝒅 𝒙 𝒊𝒔 𝒊𝒏 ℝ
1. 𝑓 𝑥 = 2𝑥
2. 𝑓 𝑥 = 10𝑥
3. 𝑓 𝑥 = 5𝑥+1
4. 𝑓 𝑥 = 32𝑥
5. 𝑓 𝑥 = 7−𝑥
6. 𝑓 𝑥 = 𝑥2
7. 𝑓 𝑥 = 1𝑥+1
8. 𝑓 𝑥 = 𝑥𝑥+1
9. 𝑓 𝑥 = (−3)
1
𝑥
10. 𝑓 𝑥 = 0−𝑥

10. Recall:
Definition:
Let 𝑎 ≠ 0. we define the following
(1) 𝑎0
= 1
(2) 𝑎−𝑛
=
1
𝑎𝑛

11. Recall
Theorem
Let r and s be rational numbers. Then,
(1) 𝑎𝑠
𝑎𝑟
= 𝑎𝑟+𝑠
(2)
𝑎𝑟
𝑎𝑠 = 𝑎𝑟−𝑠
(3) (𝑎𝑟
)𝑠
= 𝑎𝑟𝑠
(4) (𝑎𝑏)𝑟
= 𝑎𝑟
𝑏𝑟
(5)
𝑎
𝑏
𝑟
=
𝑎𝑟
𝑏𝑟

12. One-to-one Property of Exponential
Functions
If 𝑥1 ≠ 𝑥2, then 𝑏𝑥1 ≠ 𝑏𝑥2. Conversely, if
𝑏𝑥1 = 𝑏𝑥2, then 𝑥1 = 𝑥2.

13. Solve the equation 4𝑥−1
= 16
4𝑥−1
= 16
4𝑥−1 = 42
Using one-to-one property of exponential equation
𝑥 − 1 = 2
𝑥 = 1 + 2
𝑥 = 3

14. Solve the equation 125𝑥−1 = 25𝑥+3.
125𝑥−1
= 25𝑥+3
(53)𝑥−1= (52)𝑥+3
53(𝑥−1) = 52(𝑥+3)
53𝑥−3 = 52𝑥+6
Using one-to-one property of exponential function
3𝑥 − 3 = 2𝑥 + 6
3𝑥 − 2𝑥 = 6 + 3
𝑥 = 9

15. Solve the equation 9𝑥2
= 3𝑥+3
9𝑥2
= 3𝑥+3
(32
)𝑥2
= 3𝑥+3
32𝑥2
= 3𝑥+3
Using one-to-one of exponential function
2𝑥2 = 𝑥 + 3
2𝑥2
− 𝑥 − 3 = 0
2𝑥 − 3 𝑥 + 1 = 0
𝑥 =
3
2
or 𝑥 = −1

16. Try This
Solve for x.
1. 8𝑥 = 16𝑥−9
2. 11𝑥−2 = 121𝑥+3
3. 216𝑥
= 6𝑥2+2
4. 0.001𝑥 =
1
10
𝑥−5
5.
2
5
𝑥−2
=
5
2
3𝑥+2
6.
8
125
𝑥
=
4
25
𝑥+3
7.
2𝑥2
25𝑥 =
1
16
8. 3𝑥−1 = 81
9. 0.4𝑥
= 0.16𝑥−5
10. 13−𝑥+2
= 169𝑥+8