Which exponential function is represented by the table?
A. [tex]f(x)=0.2(0.5^x)[/tex]
B. [tex]f(x)=0.5(5^x)[/tex]
C. [tex]f(x)=0.5(0.2^x)[/tex]
D. [tex]f(x)=0.2(0.2^x)[/tex]
Answer (2)
Test each function with x = 0 to eliminate options.
Test the remaining functions with x = 1 to further narrow down the options.
Verify the remaining function with all the values in the table.
The exponential function represented by the table is f ( x ) = 0.5 ( 0. 2 x ) .
Explanation
Understanding the Problem We are given a table of x and f ( x ) values and asked to find which of the given exponential functions matches the table. The table is:
x f ( x ) -2 12.5 -1 2.5 0 0.5 1 0.1 2 0.02 The possible functions are:
f ( x ) = 0.2 ( 0. 5 x )
f ( x ) = 0.5 ( 5 x )
f ( x ) = 0.5 ( 0. 2 x )
f ( x ) = 0.2 ( 0. 2 x )
Testing with x=0 Let's test each function with the values from the table. We can start by testing x = 0 since it is the easiest to calculate. Recall that any number raised to the power of 0 is 1 (except 0).
f ( x ) = 0.2 ( 0. 5 x ) : f ( 0 ) = 0.2 ( 0. 5 0 ) = 0.2 ( 1 ) = 0.2 . This does not match the table since f ( 0 ) = 0.5 in the table.
f ( x ) = 0.5 ( 5 x ) : f ( 0 ) = 0.5 ( 5 0 ) = 0.5 ( 1 ) = 0.5 . This matches the table.
f ( x ) = 0.5 ( 0. 2 x ) : f ( 0 ) = 0.5 ( 0. 2 0 ) = 0.5 ( 1 ) = 0.5 . This matches the table.
f ( x ) = 0.2 ( 0. 2 x ) : f ( 0 ) = 0.2 ( 0. 2 0 ) = 0.2 ( 1 ) = 0.2 . This does not match the table since f ( 0 ) = 0.5 in the table.
So, functions 1 and 4 are not the correct functions. We are left with functions 2 and 3.
Testing with x=1 Let's test the remaining functions with x = 1 .
f ( x ) = 0.5 ( 5 x ) : f ( 1 ) = 0.5 ( 5 1 ) = 0.5 ( 5 ) = 2.5 . This does not match the table since f ( 1 ) = 0.1 in the table.
f ( x ) = 0.5 ( 0. 2 x ) : f ( 1 ) = 0.5 ( 0. 2 1 ) = 0.5 ( 0.2 ) = 0.1 . This matches the table.
Since function 2 does not match, we are left with function 3.
Final Verification Let's test function 3 with the remaining values to be sure.
f ( x ) = 0.5 ( 0. 2 x ) :
f ( 2 ) = 0.5 ( 0. 2 2 ) = 0.5 ( 0.04 ) = 0.02 . This matches the table.
f ( − 1 ) = 0.5 ( 0. 2 − 1 ) = 0.5 ( 5 ) = 2.5 . This matches the table.
f ( − 2 ) = 0.5 ( 0. 2 − 2 ) = 0.5 ( 25 ) = 12.5 . This matches the table.
Therefore, the exponential function represented by the table is f ( x ) = 0.5 ( 0. 2 x ) .
Final Answer The exponential function represented by the table is f ( x ) = 0.5 ( 0. 2 x ) .
Examples
Exponential functions are incredibly useful in modeling real-world phenomena. For example, they can describe population growth, where the rate of increase is proportional to the current population. Imagine you start with a small colony of bacteria in a petri dish. If the bacteria reproduce at a rate of 20% per hour, the exponential function f ( t ) = N 0 × ( 1 + 0.2 ) t can model the population f ( t ) after t hours, where N 0 is the initial number of bacteria. Similarly, exponential decay is used to model the decrease in the value of a car over time or the decay of radioactive substances.
The exponential function that matches the given table is f ( x ) = 0.5 ( 0. 2 x ) , confirmed by checking the values against the function. It matched at all tested values of x . Therefore, the correct answer is option C.
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