Exponential Functions

Take a look at x3 . What does it mean?

We have two parts here:
1) Exponent, which is 3.
2) Base, which is x.

x 3 = x times x times x

It's read two ways:
1) x cubed
2) x to the third power

With exponential functions, 3 is the base and x is the exponent.
So, the idea is reversed in terms of exponential functions.

Here's what exponential functions look like:

y = 3 x , f(x) = 1.124 x , etc. In other words, the exponent will be a variable.

The general exponential function looks like this: b x where the base b is ANY constant. So, the standard form for ANY exponential function is f(x) = b x where b is a real number greater than 0.

Sample: Solve for x

f(x) = 1.276 x

Here x can be ANY number we select.

Say, x = 1.2.

f(1.2) = 1.276 1.2

NOTE: You must follow your calculator's instructions in terms of exponents. Every calculator is different and thus has different steps.

I will use my TI-36 SOLAR Calculator to find an approximation for x.

f(1.2) = 1.33974088

Rounding off to two decimal places I get:

f(1.2) = 1.34

We can actually graph our point (1.2, 1.34) on the xy-plane but more on that in future exponential function lessons.

We can use the formula B(t) = 100(1.12 t ) to solve bacteria applications. We can use the above formula to find HOW MUCH bacteria remains in a given region after a certain amount of time. Of course, in the formula, lower case t = time. The number 100 indicates how many bacteria there were at the start of the LAB experiment. The decimal number 1.12 indicates how fast bacteria grows.

Sample: How much bacteria in LAB 3 after 2.9 hours of work?

Okay, t = 2.9 hours.

Replace t with 2.9 hours in the formula above and simplify.

B(2.9 hours) = 100(1.12 2.9 hours)
B(2.9 hours) = 100(1.389096016)
B(2.9 hours) = 138.9096

NOTE: An exponent can be ANY real number, positive or negative. For ANY exponential function, the domain will be ALL real numbers.

By Mr. Feliz
(c) 2005