A function of the form \$f(x) = a^x\$, where \$a > 0\$ and \$a \neq 1\$.
Note
The base \$a\$ is a positive constantand the exponent \$x\$ is a variable.
Note
The base \$a\$ must be positive and not equal to 1. Otherwise, the function is not exponential.
Graphs of Exponential Functions
Self review
1. Sketch the graph of \$f(x) = 5^x\$ and \$g(x) = \left(\frac{1}{3}\right)^x\$. 2. Identify the key properties: growth/decay, asymptotes, domain, and range.
Theory of Knowledge
How do exponential functions model real-world phenomena like population growth or radioactive decay? What are the limitations of these models?
Note
1. How do exponential functions differ from linear functions in terms of growth rate? 2. Can you think of a real-world situation where exponential decay is observed?
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