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Limits of Exponential Functions

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

Let's look at the exponential function f (x) = 4x. 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) = Cbx
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) = 2bx
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.

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