Exponential function

You already know quite a lot about numerical functions and their main properties. This time, you will go further and learn one of the most important types of numerical functions — exponential. Exponential functions are widely used in science and help model many real-life processes. You will get a general understanding of these functions, look at their plots and learn their main properties.

Exponential growth

Imagine that you have invested $1\$$ in a project. The project is very successful and brings $20\%$ of profit to you every day. In other words, your invested money is multiplied by $1.2$ every day.

It looks small at the beginning, but with such a scheme in $30$ days you will have more than $237\$$ on your account! This fast growth is called exponential. You will see it in detail in the following sections.

Exponential function

Let's start with a definition of an exponential function. It is a function of a kind:

$f(x)=a^x,a>0,a\neq1$

Here, $a$ is a constant number. It is as simple as $a$ multiplied by itself $x$ times.

Now let's return to our lucky investment example. Every day the sum is multiplied by $1.2$, so on day $x$ we are going to have $1.2^x$ dollars available, which makes our account money an exponential function of days:

$f(x)=1.2^x$

Congratulations, we have our first real (almost) example of exponential functions.

You should also know that when somebody says "exponential function", they often assume $a=e$. The number $e=2.71828\dots$ is a fundamental mathematical constant that is widely used in science. This number is irrational and has an infinite number of decimal places, so we put an ellipsis at its end. The most popular exponential function is $f(x)=e^x$, or $f(x)=exp(x)$.

Domain

Our example does not fully demonstrate an exponential function. Argument $x$ in our case is the number of days that can't be fractional or negative. However, the domain of exponential functions is $\mathbb{R}$, so $x$ can be anything.

If you struggle with negative powers, there is a way to get more intuition. We know that $2^2=4$. If we divide it by $2$, we will get the previous power:$2^1=\dfrac{2^2}{2}=2$Therefore, to get the $0$-th power and then negative powers, we need to further divide by $2$.

$2^0=\dfrac{2^1}{2}=\dfrac{2}{2}=1$$2^{-1}=\dfrac{2^0}{2}=\dfrac{1}{2}$

If we continue, we will get the formula:

$a^{-x}=\dfrac{1}{a^x}, a\neq0$

Plots

You are now ready to take a look at the plots. It will help you understand exponential functions and their properties better. In the following figure, you see two exponential functions with different bases greater than $1$.

Notice that both these functions are increasing, and the one with the larger base, $f_2$, grows faster. It is only natural because multiplication by $4$ makes the result grow faster than multiplication by $2$. The same applies to our investment example: the higher the interest rate in percent, the higher the income.

Remember that the base $a$ can be less than $1$. In the following plot, there are two examples of such functions. Look, they are very different!

Here both functions are decreasing, and the one with the smaller base, $f_4$, has a steeper plot. This is normal: since multiplying by a smaller number forces the function value to decrease faster.

Properties

Now that you have seen several examples, you can formulate the main properties of exponential functions.

• An exponential function is strictly increasing if $a>1$, and strictly decreasing if $0<a<1$.

• The domain of all exponential functions is $\mathbb{R}$.

• The codomain of all exponential functions is $\left(0,+\inf\right)$. Indeed, if your base is non-negative, you won't get a non-positive function value, no matter the power.

• Any exponential function is bounded below and unbounded above.

• Exponential functions are not symmetrical, thus being neither even nor odd.

Conclusion

In this topic, you've got better intuition behind exponential functions, and now you know that:

• An exponential function expresses the exponentiation of a non-negative constant $a$ different from 1 to the power $x$, which is an argument.

• The most popular exponential function is with base $e$.

• Exponential functions vary quite significantly depending on $a$: they are strictly decreasing when $0<a<1$ and strictly increasing when $a>1$.

Additionally, we have learned a couple of insights about negative powers.