The table above shows some values for the function f. If f(x)=ka^x for some constants k and a, what is the value of a?
The value of a is 4.
In the function f(x) = ka^x, the parameter 'a' represents the base of the exponential function. Given the values in the table, we can determine that 'a' must be 4 to satisfy the characteristics of the function as described.
Choosing 12 as the value of 'a' would imply a much steeper growth rate than what is indicated in the table. The exponential function f(x) = k(12^x) would result in values that quickly exceed those shown, thus contradicting the provided data points.
If 'a' were 14, the function f(x) = k(14^x) would again lead to an excessively rapid increase in values. This would not align with the function's behavior as depicted in the table, where the growth is more moderate compared to what would be produced with a base of 14.
Setting 'a' to 2 would result in a slower growth rate for the function f(x) = k(2^x). While this might seem plausible, the values in the table indicate that the function's growth is more pronounced than that of an exponential function with base 2.
With 'a' equal to 4, the function f(x) = k(4^x) accurately reflects the values in the table. The growth rate is consistent with the observed data points, making this choice the only viable option that aligns with the function's characteristics.
The value of 'a' in the function f(x) = ka^x is crucial for determining the growth behavior of the exponential function. Here, 'a' must be 4 to correspond correctly with the values presented in the table, ensuring that the function's output aligns with the expected growth pattern. Other options, while numerically valid, do not fit the data provided, leading to their exclusion.
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