Example Functions: f(x) = 8(3^x) and g(x) = 27(2^x)
Practice makes perfect
Since we need to find two exponential growth functions f(x) and g(x) with certain conditions, let's first review the format for these type of function. A function that models exponential growth has the following form.
y= ab^x
Here, a and b are real numbers such that a>0 and b>1. The typical form of the graph of a function like this is shown below.
As we can see from the graph above, these kind of function is increasing in its entire domain. Then, for a function f(x)=a_1b_1^x to be less than a function g(x)=a_2b_2^x for x<3, but also satisfying f(x)>g(x) for x>3, we have three conditions.
The relation for the bases is b_1>b_2. This way f(x) can surpass g(x) for x>3.
Given the first condition, it is only possible to have f(x) < g(x) for x<3, if a_2>a_1.
f(3)=g(3)
There are infinitely many functions that satisfy these conditions. We will show how to find one example. First, let's think about the values for our functions when x= 3.
f( 3)= g( 3) ⇒ a_1b_1^3 = a_2b_2^3
In particular, notice that this equality will always hold true if we choose a_1= b_2^3 and a_2= b_1^3, since we would have the identity b_2^3 b_1^3 = b_1^3 b_2^3. Furthermore, since b_1>b_2, with this choice a_2>a_1, as required. Then, we just need to choose the values for b_1 and b_2. Recall that we need to make sure that b_1>b_2.
