Overview of $e^{x}$

The basic exponential function, sometimes referred to as the exponential function, is $f(x)=e^{x}$ where $e$ is the number (approximately 2.718281828) described previously. Its graph lies between the graphs of $2^x$ and $3^x$. The graph's $y$-intercept is the point $(0,1)$, and it also contains the point $(1,e).$ Sometimes it is written as $y=\exp (x)$.

The graphs of $2^x$, $e^x$, and $3^x$

The graph of $y=e^x$ lies between that of $y=2^x$ and $y=3^x$ .

The graph of $y=e^{x}$ is upward-sloping, and increases faster as $x$ increases. The graph always lies above the $x$-axis, but gets arbitrarily close to it for negative $x$; thus, the $x$-axis is a horizontal asymptote. The graph of $e^x$ has the property that the slope of the tangent line to the graph at each point is equal to its $y$-coordinate at that point. $y=e^x$ is the only function with this property.

A Model For Proportional Change

The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change (i.e., percentage increase or decrease) in the dependent variable. If the change is positive, this is called exponential growth and if it is negative, it is called exponential decay. For example, because a radioactive substance decays at a rate proportional to the amount of the substance present, the amount of the substance present at a given time can be modeled with an exponential function. Also, because the the growth rate of a population of bacteria in a petri dish is proportional to its size, the number of bacteria in the dish at a given time can be modeled by an exponential function such as $y=Ae^{kt}$ where $A$ is the number of bacteria present initially (at time $t=0$) and $k$ is a constant called the growth constant.