Limits of Exponential Functions

Everyone has their limit; logs and exponents are no different.

Let's look at the exponential function f (x) = 4^x. No matter what value of x you throw into it, you can never get f (x) to be negative or zero. (How optimistic of it.) Try a few:

4² = 16
4³ = 64
4⁴ = 256


4⁰ = 1
4^-2 = ¹/₁₆

That's reflected in the function's graph, too. It never dips below y = 0.

Sample Problem

What's an exponential function that contains the following set of points? Hint: remember that anything to the power of 0 becomes 1.

{(2, 18), (0, 2), (3, 54)}

Because any base raised to the power of 0 equals 1, we can easily figure out the coefficient. The point (0, 2) means that y is 2 when x is 0, or f (0) = 2. Throw that bad boy into the mix and solve:

f (x) = Cb^x
2 = Cb⁰
2 = C(1)
C = 2

Now that you know C, use it to find the base. We'll use the point (2, 18), which means f (2) = 18:

f (x) = 2b^x
18 = 2b²
9 = b²

b = 3

Remember, we can't have a negative base. That's why we're only using the positive square root of 9, even though it's technically ±3.

So the exponential function we need is f (x) = 2(3)^x. Let's double-check that it works with our third point, (3, 54).

f (x) = 2(3)^x
f (3) = 2(3)³ = 2(27) = 54

Yep, we're in the clear.