To find the exponential function that represents the given data, we will analyze the values provided in the table. The data points are:
- x = 3, f(x) = 64
- x = 4, f(x) = 256
- x = 5, f(x) = 1024
We can note that the values of f(x) increase significantly as x increases by 1. We can test to see if f(x) can be represented in the form of an exponential function: f(x) = a b^(x-h), where a is a constant representing the initial value, b is the base of the exponential function, and h is the horizontal shift.
First, we rewrite the f(x) values into powers of 4:
- 64 = 4^3
- 256 = 4^4
- 1024 = 4^5
From this, we can see that:
- f(3) = 4^3
- f(4) = 4^4
- f(5) = 4^5
This indicates that for x = 3, 4, and 5, f(x) can be expressed as:
f(x) = 4^x
To finalize, we can express the exponential function in standard notation. The function that fits the data in the table is:
f(x) = 4^(x)
Thus, the exponential function representing the data in the table is f(x) = 4^x, where x represents the input values (3, 4, 5) you provided.