Exponential Functions
Exponential functions are very important type of functions used in biology,
management and social science.
Ex: The graph of exponential functions of the form 𝑦 = 𝑎 𝑥 , where 𝑎 > 1, in
particular the graph of 𝑦 = 2𝑥 is shown below. Domain: 𝑥 ∈ ℝ and Range: 𝑦 > 0.
𝑦-intercept : (0,1). As 𝑥 → ∞, 𝑦 = 𝑎 𝑥 → ∞. 𝑥-axis: Horizontal asymptote.
It is an increasing function.
Graphing of Exponential Function
The graph of exponential functions of the form 𝑦 = 𝑎 𝑥 , where 0 < 𝑎 < 1 , in
particular the graph of 𝑦 = 2−𝑥 is shown below. Domain: 𝑥 ∈ ℝ and Range: 𝑦 > 0.
𝑦-intercept : (0,1). It is a decreasing function.
Solving Exponential Equations
An equation with a variable in the exponent is called an Exponential Equations,
often can be solved using the following property:
a) Solve 25𝑥/2 = 125𝑥+3
Soln: Since bases must be the same , we write 25 as 52 and 125 as 53
25𝑥/2 = 125𝑥+3
52 𝑥/2 = 53 𝑥+3
5𝑥 = 53𝑥+9 [ 𝑎𝑚 𝑛 = 𝑎𝑚𝑛 i.e. Multiply exponent]
𝑥 = 3𝑥 + 9 [Using the above property]
2𝑥 = −9
𝑥 = −9/2
Cont…
1
b) Solve 𝑒 = 5
𝑒
1
Soln: 𝑒 𝑥 = 5 = 𝑒 −5
𝑒
𝑥
∴ 𝑥 = −5
What does 𝑒 represent?
 𝑒 is a irrational number
 𝑒 is often used as the base in an exponential equation because it provides a
good model for many practical applications
 An approximation for 𝑒 can be found on the calculator
Graph of Exponential functions
The functions 𝑦 = 2𝑥 , 𝑦 = 𝑒 𝑥 and 𝑦 = 3𝑥 are graphed for comparison.
For 𝑥 > 0, 2𝑥 < 𝑒 𝑥 < 3𝑥 .
For 𝑥 < 0, 3𝑥 < 𝑒 𝑥 < 2𝑥
All three functions has 𝑦-intercept (0,1).
Radioactive Decay
Suppose the quantity (in grams) of a radioactive substance present at time is
𝑄(𝑡) = 1000(5−0.3𝑡 ) where 𝑡 is measured in months.
a) How much will be present in 6 months?
𝑄(6) = 1000(5−0.3(6) ) = 1000(5−1.8 ) = 55.189 (g)
55.189 g will be present in 6 months
b) How long will it take to reduce the substance to 8 g?
We must find 𝑡 such that 𝑄(𝑡) = 8
i.e. 1000(5−0.3𝑡 ) = 8
5−0.3𝑡 = 8/1000 =1/125 = 5−3
−0.3𝑡 = −3
𝑡 = 10 (months)
It will take 10 months.