Which of the following functions represents exponential growth?

Which of the following functions represents exponential growth?

• A. f(x) = -2^x
• B. f(x) = 3^x
• C. f(x) = x^2
• D. f(x) = 5 - x

Answer: (B)

The function representing exponential growth is characterized by a constant base raised to a variable exponent, typically with a positive base greater than one. In this context, the function f(x) = 3^x perfectly fits this definition. As x increases, 3 raised to the power of x grows rapidly due to the nature of exponential functions; the output value doubles, triples, or otherwise increases multiplicatively rather than additively like linear functions. Therefore, as the input x becomes larger, the value of f(x) escalates dramatically, illustrating the concept of exponential growth. In contrast, the other options do not exhibit this behavior: - The first function, with a negative exponent, will yield decreasing values as x increases. - The second function, a quadratic function, exhibits parabolic growth but not exponential growth. - The last function, a linear equation, will decline as x increases, further illustrating a different mathematical behavior than what is necessary for exponential growth. In summary, the chosen function, f(x) = 3^x, is the only one that exhibits the defining characteristics of exponential growth.

The function representing exponential growth is characterized by a constant base raised to a variable exponent, typically with a positive base greater than one. In this context, the function f(x) = 3^x perfectly fits this definition.

As x increases, 3 raised to the power of x grows rapidly due to the nature of exponential functions; the output value doubles, triples, or otherwise increases multiplicatively rather than additively like linear functions. Therefore, as the input x becomes larger, the value of f(x) escalates dramatically, illustrating the concept of exponential growth.

In contrast, the other options do not exhibit this behavior:

• The first function, with a negative exponent, will yield decreasing values as x increases.

• The second function, a quadratic function, exhibits parabolic growth but not exponential growth.

• The last function, a linear equation, will decline as x increases, further illustrating a different mathematical behavior than what is necessary for exponential growth.

In summary, the chosen function, f(x) = 3^x, is the only one that exhibits the defining characteristics of exponential growth.