Question

For a start-up company to be successful, its number of clients must grown exponentially. After 3 months of operation the firm had two clients. After being operation for 5 months, the firm had 32 clients. Find an exponential function in the form of f(x)=a.bˣ

Answer 1

The exponential function modeling the growth of a company's clients based on the given data is f(x) = (1/32) · 4^x, with 'a' as 1/32 and 'b' as 4.

Step-by-step explanation:

To determine the exponential function that models the growth of a company's clients, we can use the given data points: (3, 2) and (5, 32). Here, the x-axis represents the time in months since the company started, and the y-axis represents the number of clients. The general form of an exponential function is f(x) = a · bx, where 'a' is the initial number of clients and 'b' is the base of the exponential function representing the growth factor.

We will use the given data points to solve for 'a' and 'b'.

For the first data point (3, 2), our function is 2 = a · b3. For the second data point (5, 32), the function is 32 = a · b5. By dividing the second equation by the first, we can eliminate 'a' and solve for 'b'.

32/2 = (a · b5) / (a · b3)

16 = b2 so b = 4.

Now we can use the value of 'b' to solve for 'a' using the first data point:

2 = a · 43

2 = a · 64

a = 1/32

The exponential growth function for the company's number of clients is then f(x) = (1/32) · 4x.