Exponential functions

The function that assigns to the independent variable $x$ the value of $f(x)=a^x$ is called an exponential function of base $a$, where $a$ is a positive real number other than $1$.

For example, the functions $f(x)=3^x$ and $h(x)=0.8^x$ are exponential functions of base $3$ and $0.8$ respectively.

In particular, the exponential function of base $e$, $f(x)=e^x$, is especially important since it describes the behaviour of several real situations: evolution of populations, radioactive disintegration...

Graph

The graph of the exponential function changes if its base is greater or smaller than $1$ (let's remember that it has always to be greater than zero and that it cannot be $1$).

Let's see next the graphs of $f(x)=3^x$ and $h(x)=\displaystyle \Big(\frac{1}{3}\Big)^x$ to illustrate it graphically.

It is worth mentioning that the graph of an exponential function always goes through the point $(0, 1)$.

$$f(x)=3^x$$

$$f(x)=\displaystyle \Big(\frac{1}{3}\Big)^x$$

Properties

From its graphic representation we observe that the exponential functions satisfies the following properties:

• Domain: $Dom (f) = \mathbb{R}$
• Image: $Im (f) = (0, +\infty)$
• Bounds: bounded from below by $0$
• Intersection with the axes: It cuts with the vertical axis at $y= 1$. It does not cut the horizontal axis.
• Continuity: It is continuous in $\mathbb{R}$
• Asimptotes: The straight line $y= 0$ is a horizontal asimptote (but only in one of the extremes)
• Regularity: It is not periodic.
• Symmetries: It is not symmetric.
• Monotonicity: If $a> 1$, the function is strictly increasing. If $a <1$, the function is strictly decreasing.
• Relative extrema: It does not have any.
• Injectivity and exhaustivity: It is injective (the images of different points are different), but it is not exhaustive since the image is not $\mathbb{R}$