Exponential functions

The exponential function is of the form

where $a$ and $b$ are parameters. $a$ is (as always) called the base, and is a positive number

The value for $b$ is a real number different from zero. It is important to note that the input $x$ is in the exponent, and this gives this function its name. So,

is an exponential function (with base is $3$), while

is not, because the input $x$ is now in the base as is called a power function.

The graph of the two exponential functions

and

are shown below:

From the graphs above, we can observe some features which are generally true for exponential functions $f(x)=b a^x$:

1. The graph of an exponential function approaches but never touches the $x$-axis. The reason is that $a^x$ is never $0$, whatever number we choose for $x$. Thus, exponential functions have no $x$-intercept.
2. The $y$-intercept is $b$, as $f(0)=b\cdot a^0 =b\cdot 1=b$.
3. For a base bigger than $1$, the graph grows from left to right, while for a base between $0$ and $1$ the graph falls from left to right.
4. For every step of length $1$ to the right does the height of the graph ($y$) change by a factor $a$. For the example above, $f(x)=3^x$, at $x=0$ it is $y=1$, and at $x=1$ the height $y$ is three times bigger, and at $x=2$ again three times bigger, and so on. You can also inspect the table of values to see this behaviour.

Is a function of the form

also an exponential function? Indeed it is, because we write

In fact, every function of the form

is an exponential function.