Basic Exponential Functions

Thinking about formulas, basic exponential functions are functions whose formulas look like

Just a leading coefficent and a base greater than \(1\).

Here is the graph of \(y = 2^x\).

The domain of a basic exponential function is all real numbers, \((-\infty , \infty )\) and it increases on this domain.

Graphically, the horizontal axis is a horizontal asymptote.

\[ \lim _{x \to -\infty } 2^x = \answer {0} \, \text { and } \, \lim _{x \to \infty } 2^x = \answer {\infty } \]

Note:. Using the number \(e\) as the base of our basic exponential function is very popular.

The roles are reversed when the base is less than \(1\)

Here is the graph of \(y = \left (\frac {1}{3}\right )^x\).

When the base is less than \(1\), then the whole function decreases. The function is still unbounded to one side and approaches \(0\) on the other.

\[ \lim _{x \to -\infty } \left (\frac {1}{3}\right )^x = \answer {\infty } \, \text { and } \, \lim _{x \to \infty } \left (\frac {1}{3}\right )^x = \answer {0} \]

Note that

The function \(f(x) = \left (\frac {1}{3}\right )^x\) can be written as

We can view any exponential function with a base less than \(1\) as an exponential function with a base greater than \(1\) and just change the sign of the exponent.

Here is the graph of \(y = \left ( 3 \right )^{-x}\).

Instead of using numbers greater or less than \(1\), we could always use a base graeter than \(1\) and just use negative exponents. In that case, we might as well use \(e\) as our base.

This gives us a basic basic exponential function: \(e^x\)

And, then three other alternative choices, if you prefer.

Basic exponential functions become unbounded in one direction, while approaching \(0\) in the other direction.

• When the base is greater than \(1\), exponential functions tend to \(0\) in the direction that makes their exponent negative.
  
• When the base is less than \(1\), exponential functions tend to \(0\) in the direction that makes their exponent positive.
  

Expoential functions are unbounded in the other direction.

Exponential funcitons can be unbounded positively or negatively. This is determined by the sign of the leading coefficient.

Those are our basic exponential functions. Pick one as your own basic exponential function.

\(exp(x) = e^x\) is very popular.

Then compare all other exponential functions to it.

General Exponential Functions

Our general template for exponential functions looks like

Of we choose \(e\) as the base, then they look like

• \(A\) is the leading coefficent for the function.
• \(B\) is the leading coefficent of the exponent.

Comparing these back to our basic exponential functions, we get

• \(A > 0\) and \(B > 0\) gives an increasing exponential function.
• \(A < 0\) and \(B > 0\) gives a decreasing exponential function.
• \(A > 0\) and \(B < 0\) gives a decreasing exponential function.
• \(A < 0\) and \(B < 0\) gives an increasing exponential function.

\(\blacktriangleright \) When the leading coefficients are the same sign, then the exponential function is increasing.

\(\blacktriangleright \) When the leading coefficients are different signs, then the exponential function is decreasing.

Graphically

Even though the graphs of exponential functions appear to increase quite quickly, there are no vertical asymptotes. The domains include all real numbers. The function just increases or decreases very quickly and continues to do so.

Here is the graph of \(f(x) = -\left ( 3 \right )^{-x + 1}\).

This matches our template for an exponential function.

• The leading coefficient is \(-1\), which is negative.
• The leading coefficient of the exponent is \(-1\), which is negative.

Both leading coefficients have the same sign, so this exponential function is increasing.

The leading coefficient of the function is \(-1\), which is negative. Therefore, the function only has negative values.

The exponent is \(-x-5\). This is negative in the positive tail of the domain. The function tends to \(0\) in this direction. The function becomes unbounded in the other direction. Unbounded negatively.

\[ \lim \limits _{x \to \infty } f(x) = 0 \]

\[ \lim \limits _{x \to -\infty } f(x) = -\infty \]