- Identify the base of the exponential function, which is \(b = \frac{3}{2} = 1.5\).
- Apply the classification rule: if \(b > 1\), the function represents exponential growth.
- Since \(1.5 > 1\), the function \(h(x) = \left(\frac{3}{2}\right)^x - 1\) is an exponential growth function.
- The final answer is \(\boxed{\text{growth}}\).
Explanation
- Analyzing the Function Structure
The given function is \(h(x) = \left(\frac{3}{2}\right)^x - 1\). An exponential function is typically written in the form \(f(x) = a \cdot b^x + c\), where \(b\) is the base, and \(a\) and \(c\) are constants. In this specific function, the base is \(b = \frac{3}{2}\). To determine whether this function represents exponential growth or exponential decay, we examine the value of the base \(b\).
- Classification Criteria for Exponential Functions
An exponential function \(f(x) = b^x\) is classified based on its base \(b\):
- If \(b > 1\), the function represents exponential growth, because as \(x\) increases, \(f(x)\) increases at an accelerating rate.
- If \(0 < b < 1\), the function represents exponential decay, because as \(x\) increases, \(f(x)\) decreases towards zero.
In our case, the base is \(b = \frac{3}{2} = 1.5\). Since \(1.5 > 1\), the exponential component \(\left(\frac{3}{2}\right)^x\) will increase as \(x\) increases.
- Conclusion on the Function Type
Comparing the base \(b = 1.5\) to the value 1, we clearly see that \(1.5 > 1\). Because the base is greater than 1, the function \(h(x) = \left(\frac{3}{2}\right)^x - 1\) exhibits exponential growth behavior regardless of the constant shift \(-1\). Therefore, the function is an exponential growth function.
Examples
Exponential growth is a common concept in finance and biology. For instance, if you invest money in a savings account with a compound interest rate, your balance grows exponentially over time. Similarly, in biology, a population of bacteria under ideal conditions will multiply at a constant percentage rate, resulting in an exponential growth pattern. Identifying whether a model represents growth or decay is crucial for predicting future outcomes in these real-world scenarios.