Exponential Functions

Definition

Exponential functions are mathematical expressions of the form $$f(x) = a imes b^{x}$$, where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent. These functions are characterized by their rapid growth or decay, depending on whether the base is greater than one or between zero and one. They play a crucial role in various real-world applications, including population growth, radioactive decay, and finance.

5 Must Know Facts For Your Next Test

1. Exponential functions can model situations where quantities grow or shrink at a rate proportional to their current value, such as populations or investments.
2. The graph of an exponential function is always either increasing or decreasing but never flat; this characteristic curve approaches but never touches the x-axis.
3. For a base greater than 1, exponential functions exhibit exponential growth, while for a base between 0 and 1, they exhibit exponential decay.
4. Exponential functions are continuous and defined for all real numbers, making them versatile in mathematical modeling.
5. The function's horizontal asymptote is at $$y = 0$$; this means that as x approaches negative infinity, the value of the function approaches zero.

Review Questions

• How do you identify whether an exponential function represents growth or decay?
  • To determine if an exponential function represents growth or decay, look at the base 'b' in the function $$f(x) = a imes b^{x}$$. If the base 'b' is greater than 1, the function will exhibit exponential growth as x increases. Conversely, if the base 'b' is between 0 and 1, it indicates exponential decay since the function's value decreases as x increases.
• What role do exponential functions play in real-world applications such as finance and biology?
  • Exponential functions are vital in modeling various real-world scenarios. In finance, they help calculate compound interest, allowing investors to understand how their investments grow over time. In biology, these functions describe population growth, where organisms reproduce at rates proportional to their current population size. This makes exponential functions essential for predicting trends and behaviors in diverse fields.
• Evaluate how changing the base of an exponential function affects its graph and growth rate.
  • Changing the base of an exponential function significantly impacts its graph and growth rate. A larger base results in a steeper graph with faster growth as x increases, while a smaller base (greater than 1 but less than the original) creates a more gradual slope. When using bases less than 1, it flips to decay, making it decrease faster with smaller values. Therefore, the choice of base directly influences not only how quickly values change but also how the function behaves overall.

Related terms

Base: In exponential functions, the base is the constant value raised to the power of the variable. It determines the rate of growth or decay.

Logarithm: The logarithm is the inverse operation of exponentiation, helping to solve equations involving exponential functions by expressing them in terms of their bases.

Growth Rate: The growth rate is a key feature of exponential functions that describes how quickly the function's value increases over time when the base is greater than one.