3.3 Exponential Functions

You may have heard of exponential functions before, but what are they? Put simply, an exponential function is a function of some positive real number that is not equal to 1, raised to the power of a variable exponent. Formally, an exponential function with base $a$ is a function of the form

$f(x)=a^x$

where $a$ and $x$ are real numbers, and $a>0$ and $a\ne 1$. The reason that 1 is ruled out is that any power of 1 is 1, which means that $f(x)$ becomes constant as $f(x)=1^x=1$. The domain of $f(x)=a^x$ for $a>0$ and $a\ne 1$ is the set of all real numbers. An exponential function can describe growth or decay.

For example, say you can invest in an account that will grow your investment by 5% annually. In the real world, you will receive interest on not only your original investment but also on interest or dividends you have earned. In one year, your investment will be 1.05 times or 105% of the original investment. In two years, your investment will be 1.1025 times or 110.25% of the original, and so on. Following the pattern, in $x$ years, you will have ${1.05}^x$ times the original investment.

To build a solid foundation, let’s first practice evaluating exponential functions. Suppose there are three functions:

$\begin{array}{rl}& f(x)=3^x\\ & g(x)=2^{5-x}\\ & h(x)=5^{-x}\end{array}$

Evaluate each function at $x=3$ for $f(x)$, at $x=-4$ for $g(x)$, and at $x=2$ for $h(x)$.

What we need to do is plug $x$ into each function and find the outcome of the function.

$\begin{array}{rl}& f(x=3)=3^3=27\\ & g(x=-4)=2^{5-(-4)}=2^9=512\\ & h(x=2)=5^{-2}={\left(\frac{1}{5}\right)}^2=\frac{1}{25}\end{array}$

Now, let’s go back to the investment example. If you can grow your investment at a rate of 5%, how much will your investment be worth in five years? In order to solve this, we will evaluate the function as follows:

$f(x=5)={1.05}^5=1.2762815625$

This tells us that the value of the investment in five years will be 1.2763 times or 127.63% of the original invesment. How about in 30 years?

$f(x=30)={1.05}^{30}=4.32194237515$

So it will be 432.19% of the original investment. If you invest $100, then you will have $432.19 in the account in 30 years.

Do you remember the nine exponent rules from the previous section? If not, go back and practice those properties before moving on.