Practice Exponential functions
An exponential function models a constant multiplication factor over equal steps, rather than a constant added amount.
It describes growth and decay such as interest or cooling models.
Exercises with solutions
An exponential function has the form f(x) = a · b^x. The starting value a is the value at x = 0. The growth factor b says how the value changes at each step: b > 1 means increase, 0 < b < 1 decrease. Unlike linear functions, the value changes by the same factor at each step, not by the same amount.
Exercise 1
A bacteria culture starts with 200 bacteria and doubles every hour. What is the function?
Show solution and working
f(x) = 200 · 2^x
The starting value is 200. The growth factor is 2 because the count doubles every hour.
Exercise 2
How many bacteria are there after 3 hours?
Show solution and working
1,600
f(3) = 200 · 2³ = 200 · 8 = 1,600.
Exercise 3
A car loses 20 % of its value each year. What is the growth factor?
Show solution and working
0.8
With a decrease of 20 %, 80 % remains: b = 1 − 0.2 = 0.8.
Exercise 4
A capital of €1,000 grows by 5 % per year. How much is it after 2 years?
Show solution and working
€1,102.50
f(2) = 1,000 · 1.05² = 1,000 · 1.1025 = 1,102.50.
Common mistake
Simply adding for percentage growth. 5 % over two years does not give 10 % but the factor 1.05² = 1.1025, so 10.25 %. And swapping the starting value and the growth factor.
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