What exponential function fits the data points in the table where x=3 gives f(x)=64, x=4 gives f(x)=256, and x=5 gives f(x)=1024?

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Finding the Exponential Function

To determine the exponential function that fits the given data points, we can start by analyzing the values of f(x) and x provided in the table:

  • x = 3, f(x) = 64
  • x = 4, f(x) = 256
  • x = 5, f(x) = 1024

From this data, we observe that:

  • 64 = 4^3
  • 256 = 4^4
  • 1024 = 4^5

This indicates that each value of f(x) can be expressed as a power of 4.

To derive the general form of the exponential function, we can express f(x) as:

f(x) = 4^x

Next, we can confirm this function by checking if it produces the correct outputs for the given inputs:

  • For x = 3: f(3) = 4^3 = 64
  • For x = 4: f(4) = 4^4 = 256
  • For x = 5: f(5) = 4^5 = 1024

Since the outputs match the values in your table, we can conclude that the exponential function representing your data is:

Final Function

f(x) = 4^x

This function accurately represents the relationship between x and f(x) as observed in the table provided.

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