Introduction to Exponential Functions

Learning Objectives

• Graph exponential equations and functions.
• Solve applied problems using exponential problems and their graphs.

In addition to linear, quadratic, rational, and radical functions, there are exponential functionsA function of the form `f(x)=b^x` where `b>0` and `b!=1`.. Exponential functions have the form `f(x) = b^x`, where `b>0` and `b!=1`. Just as in any exponential expression, `b` is called the baseThe expression that is being raised to a power when using exponential notation. In `5^3`, `5` is the base and is the number that is repeatedly multiplied. `5^3=5*5*5`. And in `a^b`, the base is `a`. and `x` is called the exponentWhen a number is expressed in the form `a^b`, `b` is the exponent. The exponent indicates how many times the base is used as a factor. Power and exponent mean the same thing. .

An example of an exponential function is the growth of bacteria. Some bacteria double every hour. If you start with `1` bacterium and it doubles every hour, you will have `2^x` bacteria after `x` hours. This can be written as `f(x) = 2^x`.

Before you start, `f(0) = 2^0 = 1`

After `1` hour, `f(1) = 2^1 = 2`

In `2` hours, `f(2) = 2^2 = 4`

In `3` hours, `f(3) = 2^3 = 8`

and so on.

With the definition `f(x) = b^x` and the restrictions that `b>0` and that `b!=1`, the domain of an exponential function is the set of all real numbers. The range is the set of all positive real numbers. The following graph shows `f(x) = 2^x`.

As you can see above, this exponential function has a graph that gets very close to the `x`-axis as the graph extends to the left (as `x` becomes more negative), but never really touches the `x`-axis. Knowing the general shape of the graphs of exponential functions is helpful for graphing specific exponential equations or functions.

Making a table of values is also helpful, because you can use the table to place the curve of the graph more accurately. One thing to remember is that if a base has a negative exponent, then take the reciprocal of the base to make the exponent positive. For example, `4^-2=(4/1)^-2=(1/4)^2`. 

Look at the table of values. Think about what happens as the `x` values increase: so do the function values (`f(x)` or `y`)!

Now that you have a table of values, you can use these values to help you draw both the shape and location of the function. Connect the points as best you can to make a smooth curve (not a series of straight lines). This shows that all of the points on the curve are part of this function.

This is an example of exponential growthAn exponential function of the form `f(x)=b^x`, where `b>1` and `b!=0`. The function increases as `x` increases. . As `x` increases, `f(x)` “grows” more quickly. Let’s try another one.

Let’s compare the three graphs you’ve seen. The functions `f(x) = 2^x`, `f(x) = 3^x`, and `f(x) = 4^x` are all graphed below.

Notice that a larger base makes the graph steeper. A larger base also makes the graph closer to the `y`-axis for `x>0` and closer to the `x`-axis for `x<0`. All the graphs go through `(0, 1)`!

Remember that for exponential functions, `b>0` , but `b!=1`. In the examples above, `b>1`. What happens when `b` is between `0` and `1`, `0<b<1`?

Notice that the shape is similar to the shape when `b>1`, but this time the graph gets closer to the `x`-axis when `x>0`, rather than when `x<0`. This is exponential decayAn exponential function of the form `f(x) = b^x`, where `0 < b < 1`. The function decreases as `x` increases.. Instead of the function values “growing” as values increase, as they did before, the function values “decay” or decrease as `x` values increase. They get closer and closer to `0`.

Exponential functions can be used in many contexts, such as compound interest (money), population growth, and radioactive decay. In most of these, however, the function is not exactly of the form `f(x) = b^x`. Often, this is adjusted by adding or multiplying constants.

For example, the compound interest formula is `A=P(1+r/m)^(mt)`, where `P` is the principalIn finance, the amount of money on which interest is calculated. (the initial investment that is gathering interest) and `A` is the amount of money you would have, with interest, at the end of `t` years, using an annual interest rate of `r` (expressed as a decimal) and `m` compounding periods per year. In this case, the base is the value represented by the expression `1+r/m` and the exponent is `mt`, a product of two values.

Radioactive decay is an example of exponential decay. Radioactive elements have a half-lifeThe amount of time it takes a substance to decrease to half its original amount.. This is the amount of time it takes for half of a mass of the element to decay into another substance. For example, uranium-`238` is a slowly decaying radioactive element with a half-life of about `4.47` billion years. That means it will take that long for `100` grams of uranium-`238` to turn into `50` grams of uranium-`238` (the other `50` grams will have turned into another element). That’s a long time! On the other extreme, radon-`220` has a half-life of about `56` seconds. What does this mean? `100` grams of radon-`220` will turn into `50` grams of radon-`220` and `50` grams of something else in less than a minute!

Since the amount is halved each half-life, an exponential function can be used to describe the amount remaining over time. The formula `R=A(1/2)^(t/h)` gives the remaining amount `R` from an initial amount `A`, where `h` is the half-life of the element and `t` is the amount of time passed (using the same time unit as the half-life).

Exponential functions of the form `f(x) = b^x` appear in different contexts, including finance and radioactive decay. The base `b` must be a positive number and cannot be `1`. The graphs of these functions are curves that increase (from left to right) if `b>1`, showing exponential growth, and decrease if `0<b<1`, showing exponential decay.