Which of the following represents an exponential function?

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Accepted Answer

f(x) = 2^x. An exponential function has the form f(x) = a·b^x, where the variable x is in the exponent and b is a positive constant (b > 0, b ≠ 1). The function f(x) = 2^x fits this form with a = 1 and b = 2. Exponential functions are characterized by rapid growth (when b > 1) or rapid decay (when 0 < b < 1). The graph of 2^x increases rapidly to the right and approaches 0 (but never reaches it) to the left, with the x-axis serving as a horizontal asymptote. "f(x) = 3x + 2", f(x) = 3x + 2, is a linear function (the highest power of x is 1). "f(x) = x^3 + 1", f(x) = x^3 + 1, is a polynomial function (specifically a cubic function) - the variable x is the base and the exponent is a constant, which is the opposite of an exponential function. This is a critical distinction: in x^3, the base varies and the exponent is fixed; in 2^x, the base is fixed and the exponent varies. "f(x) = √x", f(x) = √x = x^(1/2), is a radical (or power) function. Recognizing different function types - linear, quadratic, polynomial, exponential, logarithmic, radical, and rational - is an essential skill for the CLEP exam. Exponential functions model many real-world phenomena including population growth, radioactive decay, compound interest, and bacterial growth. Their key property is that they change by a constant multiplicative factor over equal intervals, rather than by a constant additive amount (which characterizes linear functions).

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