Complete the tables of values.
[tex]\begin{tabular}{|c|c|}\hline$x$ & $4^{-x}$ \\\hline-1 & 4 \\\hline 0 & $a$ \\\hline 2 & $b$ \\\hline 4 & $c$ \\\hline\end{tabular}[/tex]
[tex]\begin{tabular}{|c|c|}\hline$x$ & $\left(\frac{2}{3}\right)^x$ \\\hline-1 & $\frac{3}{2}$ \\\hline 0 & $d$ \\\hline 2 & $e$ \\\hline 4 & $f$ \\\hline\end{tabular}[/tex]
[tex]$a=\square b=\square$[/tex]
[tex]$c=\square$[/tex]
[tex]$d=\square e=\square$[/tex]
[tex]$f=\square$[/tex]
Answer (2)
Calculate a by substituting x = 0 into 4 − x , which gives a = 4 − 0 = 1 .
Calculate b by substituting x = 2 into 4 − x , which gives b = 4 − 2 = 16 1 .
Calculate c by substituting x = 4 into 4 − x , which gives c = 4 − 4 = 256 1 .
Calculate d by substituting x = 0 into ( 3 2 ) x , which gives d = ( 3 2 ) 0 = 1 .
Calculate e by substituting x = 2 into ( 3 2 ) x , which gives e = ( 3 2 ) 2 = 9 4 .
Calculate f by substituting x = 4 into ( 3 2 ) x , which gives f = ( 3 2 ) 4 = 81 16 .
The final answers are: a = 1 , b = 16 1 , c = 256 1 and d = 1 , e = 9 4 , f = 81 16 .
Explanation
Understanding the Problem We are given two tables with functions and need to complete them by finding the values of a , b , c , d , e , and f . We will substitute the given x values into the corresponding functions to find these values.
Calculating a To find a , we substitute x = 0 into the function 4 − x . Thus, we have a = 4 − 0 = 1 .
Calculating b To find b , we substitute x = 2 into the function 4 − x . Thus, we have b = 4 − 2 = 4 2 1 = 16 1 = 0.0625 .
Calculating c To find c , we substitute x = 4 into the function 4 − x . Thus, we have c = 4 − 4 = 4 4 1 = 256 1 = 0.00390625 .
Calculating d To find d , we substitute x = 0 into the function ( 3 2 ) x . Thus, we have d = ( 3 2 ) 0 = 1 .
Calculating e To find e , we substitute x = 2 into the function ( 3 2 ) x . Thus, we have e = ( 3 2 ) 2 = 3 2 2 2 = 9 4 = 0.444444... .
Calculating f To find f , we substitute x = 4 into the function ( 3 2 ) x . Thus, we have f = ( 3 2 ) 4 = 3 4 2 4 = 81 16 = 0.197530864... .
Final Answer Therefore, the completed values are: a = 1 , b = 16 1 , c = 256 1 , d = 1 , e = 9 4 , and f = 81 16 .
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, the decay of a radioactive substance can be modeled by an exponential function of the form N ( t ) = N 0 e − k t , where N ( t ) is the amount of the substance remaining after time t , N 0 is the initial amount, k is the decay constant, and e is the base of the natural logarithm. Understanding how to evaluate exponential functions is crucial for making predictions and analyzing data in these contexts.
We calculated the values for each variable by substituting the given x values into the expressions. The final results are a = 1 , b = 16 1 , c = 256 1 , d = 1 , e = 9 4 , and f = 81 16 .
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