Analyzing Exponential Function F(x) = 3 * 2^x Properties And Characteristics

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In the realm of mathematics, exponential functions hold a significant position due to their unique properties and widespread applications. These functions, characterized by a constant base raised to a variable exponent, exhibit fascinating behaviors that are crucial to understanding various real-world phenomena. This article delves into the intricacies of a specific exponential function, f(x) = 3 * 2^x, meticulously examining its domain, y-intercept, and monotonicity to determine the veracity of several statements. By employing a combination of analytical techniques and graphical representations, we aim to provide a comprehensive understanding of this function and its characteristics.

Decoding Exponential Functions: An In-Depth Exploration

To embark on our exploration of f(x) = 3 * 2^x, it is imperative to first grasp the fundamental nature of exponential functions. In general, an exponential function takes the form f(x) = a * b^x, where 'a' represents the initial value, 'b' denotes the base (a positive constant not equal to 1), and 'x' signifies the exponent. The base 'b' plays a pivotal role in dictating the function's behavior. When 'b' is greater than 1, the function exhibits exponential growth, while values of 'b' between 0 and 1 lead to exponential decay.

Now, let us turn our attention to our specific function, f(x) = 3 * 2^x. Here, the initial value 'a' is 3, and the base 'b' is 2. Since the base is greater than 1, we can anticipate that this function will demonstrate exponential growth. The coefficient 3 acts as a vertical stretch, amplifying the function's values compared to the basic exponential function 2^x. This foundational understanding will serve as a bedrock for our subsequent analysis of the statements provided.

Statement A: Delving into the Domain of f(x) = 3 * 2^x

The domain of a function encompasses all possible input values (x-values) for which the function produces a valid output. To ascertain the domain of f(x) = 3 * 2^x, we must consider any restrictions on the input values. In the case of exponential functions, there are no inherent restrictions on the exponent 'x'. We can raise 2 to any real number, whether positive, negative, or zero, and the result will always be a real number. Consequently, multiplying this result by 3 will also yield a real number.

Therefore, the domain of f(x) = 3 * 2^x extends across all real numbers. This can be mathematically expressed as (-∞, ∞). The statement that the domain is x > 0 is incorrect. Exponential functions, unlike some other function types, do not have domain restrictions imposed by operations such as division by zero or taking the square root of a negative number. The exponent can freely take any real value, making the domain all real numbers.

Statement B and D: Unveiling the Y-Intercept of f(x) = 3 * 2^x

The y-intercept of a function represents the point where the function's graph intersects the y-axis. This intersection occurs when the input value, x, is equal to 0. To determine the y-intercept of f(x) = 3 * 2^x, we simply substitute x = 0 into the function's equation:

f(0) = 3 * 2^0

Since any non-zero number raised to the power of 0 equals 1, we have:

f(0) = 3 * 1 = 3

Thus, the y-intercept of f(x) = 3 * 2^x is the point (0, 3). Statement B, which claims the y-intercept is (0, 1), is incorrect. Statement D, asserting the y-intercept is (0, 3), is correct. The y-intercept is a crucial characteristic of a function, providing a fixed point on the graph and serving as a reference for understanding the function's behavior as x varies.

Statement C: Examining the Monotonicity of f(x) = 3 * 2^x

Monotonicity refers to the property of a function either consistently increasing or consistently decreasing over its entire domain. To analyze the monotonicity of f(x) = 3 * 2^x, we must consider its derivative. The derivative of a function provides insights into its rate of change. A positive derivative indicates an increasing function, while a negative derivative signifies a decreasing function.

The derivative of f(x) = 3 * 2^x can be found using the rules of calculus. The derivative of a constant multiplied by a function is simply the constant multiplied by the derivative of the function. The derivative of 2^x is 2^x * ln(2), where ln(2) represents the natural logarithm of 2. Therefore, the derivative of f(x) is:

f'(x) = 3 * 2^x * ln(2)

Since 2^x is always positive for any real number x, and ln(2) is also a positive constant (approximately 0.693), the derivative f'(x) is always positive for all values of x. This positive derivative definitively indicates that f(x) = 3 * 2^x is an increasing function over its entire domain. Statement C, which asserts that the function is always decreasing, is therefore incorrect. The exponential growth nature of the function, stemming from its base being greater than 1, ensures its consistent increase as x increases.

Conclusion: Unraveling the Truth about f(x) = 3 * 2^x

In conclusion, our comprehensive analysis of f(x) = 3 * 2^x has revealed that only statement D, asserting the y-intercept is (0, 3), is true. We have demonstrated that the domain of the function encompasses all real numbers, contradicting statement A. Furthermore, we have established that the function is always increasing, invalidating statement C. By meticulously examining the function's properties, including its domain, y-intercept, and monotonicity, we have gained a deeper understanding of its behavior and characteristics.

This exploration underscores the importance of a thorough understanding of exponential functions in mathematics. Their unique properties and behaviors make them invaluable tools for modeling and analyzing a wide range of phenomena in various fields, including finance, biology, and physics. By mastering the concepts presented in this article, readers can confidently tackle problems involving exponential functions and apply their knowledge to real-world scenarios.

Key Takeaways

  • Exponential functions are of the form f(x) = a * b^x, where 'a' is the initial value and 'b' is the base.
  • The domain of f(x) = 3 * 2^x is all real numbers.
  • The y-intercept of f(x) = 3 * 2^x is (0, 3).
  • f(x) = 3 * 2^x is an increasing function over its entire domain.