Evaluate the function for the given value.
[tex]
\begin{array}{c}
y=4 \cdot 3^x \text { for } x=3 \\
y=[?]
\end{array}
[/tex]
Answer (2)
Substitute x = 3 into the function y = 4 c d o t 3 x .
Calculate 3 3 = 27 .
Multiply the result by 4: 4 c d o t 27 = 108 .
The final answer is 108 .
Explanation
Understanding the problem We are given the function y = 4 c d o t 3 x and we need to evaluate it at x = 3 . This means we need to substitute x = 3 into the expression and calculate the value of y .
Substituting the value of x First, substitute x = 3 into the function: y = 4 c d o t 3 3
Calculating the exponent Next, we calculate 3 3 , which is 3 c d o t 3 c d o t 3 = 27 . So we have: y = 4 c d o t 27
Final Calculation Finally, we multiply 4 by 27 to find the value of y : y = 4 c d o t 27 = 108
Examples
Exponential functions like y = 4 c d o t 3 x are used to model various real-world phenomena, such as population growth, compound interest, and radioactive decay. For example, if a bacterial population triples every hour and you start with 4 bacteria, then after 3 hours, you would have 4 c d o t 3 3 = 108 bacteria. Understanding how to evaluate these functions helps in predicting future values in these scenarios.
To evaluate the function y = 4 ⋅ 3 x at x = 3 , substitute 3 into the equation to get y = 4 ⋅ 3 3 . This simplifies to y = 4 ⋅ 27 , resulting in y = 108 . Therefore, the value of y is 108 .
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