# Is f(x) = \frac{6x}{2^x} a linear exponential or power function? What is the difference between...

## Question:

Is {eq}f(x) = \frac{6x}{2^x} {/eq} a linear exponential or power function? What is the difference between these functions?

## Functions:

Recall that a linear function is so-called because it describes a line; we can always write one in the usual form {eq}y = mx + b {/eq}. An exponential function is one where the variable is the exponential; they are typically of the form {eq}y = a^x {/eq}. A power function is one where the variable is being raised to a specific power; they usually look like {eq}y = x^a {/eq}.

This function is not of the form {eq}y = mx + b {/eq}, so it is obviously not linear (it just has a linear element to it). We have no power functions whatsoever; the dominating piece of this function is {eq}2^x {/eq}, which is an exponential function since the variable is an exponent. The given function is an exponential function.

Linear functions are clearly different from any other: they are lines; exponential and power functions are not. The difference between a power function and an exponential function is in whether the variable is the exponent or the base. If it is the base, then the exponent is constant, and we call it a power, so {eq}y = x^a {/eq} is a power function. If the variable is the exponent, then the function is exponential (since now the exponent changes) and has the elementary form {eq}y = a^x {/eq}.