What is an exponential function?

An exponential function is defined as a function in which the variable appears in the exponent. The correct form of an exponential function is expressed as ( f(x) = a(b^x) ), where ( a ) is a non-zero constant that acts as a scaling factor, and ( b ) is the base of the exponential expression which is a positive constant. This definition captures the essence of exponential growth or decay, where the rate of change of the function is proportional to its current value.

In this form, as ( x ) increases or decreases, the function rapidly changes, leading to characteristic curves that rise or fall steeply, distinguishing them from linear or polynomial functions. This highlights the unique property of exponential functions where the base raised to a power represents continuous growth or decrease, essential in various applications such as population dynamics, finance, and natural processes.

While other options define different types of mathematical functions, they do not fit the criteria for an exponential function. For instance, linear functions (the first option) have constant rates of change, parabolic functions (the third option) describe quadratic relationships, and logarithmic functions (the fourth option) are the inverses of exponential functions rather than being exponential themselves.