When evaluating the range of a function, it is crucial to understand its behavior and growth pattern. In the case of the function f(x) = 2(3)^x, we are dealing with an exponential function where the base is 3 and the coefficient is 2. This function represents exponential growth, where the value of x increases, the value of the function grows exponentially. By delving deeper into the understanding of this function, we can analyze its range and how it behaves for different values of x.
Understanding the Exponential Growth of f(x) = 2(3)^x
Exponential growth is a phenomenon where the rate of increase of a quantity is proportional to its current value. In the function f(x) = 2(3)^x, the base 3 signifies the rate of growth, which is exponential. As x increases, the value of 3^x grows rapidly, and when multiplied by 2, the function f(x) experiences even faster growth. This exponential growth pattern is characteristic of functions with a base greater than 1, leading to unbounded growth as x approaches infinity. Understanding this fundamental concept is essential in evaluating the range of f(x) = 2(3)^x.
To visualize the exponential growth of f(x) = 2(3)^x, consider plotting the function on a graph. As x increases, the function grows rapidly, showing an upward trend that approaches infinity. The steepness of the curve is a reflection of the exponential growth rate dictated by the base 3. This growth pattern highlights the significance of the base in determining the behavior of the function. By understanding the exponential growth of f(x) = 2(3)^x, we can anticipate the range of values the function can take as x varies.
Analyzing the Behavior of f(x) for Various Values of x
Analyzing the behavior of f(x) for different values of x allows us to determine the range of the function. As x approaches negative infinity, the value of 3^x decreases exponentially, leading to values close to zero. When multiplied by 2, the function f(x) approaches 0 but never reaches it. As x increases, the function grows rapidly towards infinity, showcasing the unbounded nature of exponential growth. By examining the behavior of f(x) for various values of x, we can identify the range of values the function can attain, from close to zero to approaching infinity.
In conclusion, evaluating the range of f(x) = 2(3)^x requires a thorough understanding of exponential growth and the behavior of the function for different values of x. The exponential nature of the function, characterized by a base of 3, signifies rapid and unbounded growth as x increases. By analyzing the behavior of f(x) on a graph and for various values of x, we can grasp the range of values the function can take. Understanding the exponential growth of f(x) = 2(3)^x provides valuable insights into its behavior and helps in determining its range effectively.
