Evaluate the function for the given value.
[tex]
\begin{array}{c}
y=6 \cdot 2^x \text { for } x=1 \\
y=[?]
\end{array}
[/tex]
Answer (2)
Substitute x = 1 into the function y = 6 × 2 x .
Calculate 2 1 = 2 .
Multiply 6 × 2 to get 12 .
The value of the function at x = 1 is 12 .
Explanation
Understanding the problem We are given the function y = 6 × 2 x and we want to evaluate it at x = 1 . This means we need to substitute x = 1 into the expression for y .
Substitution Substitute x = 1 into the function: y = 6 × 2 1
Simplification Now, we simplify the expression. We know that 2 1 = 2 , so we have: y = 6 × 2 = 12
Final Answer Therefore, when x = 1 , the value of the function is y = 12 .
Examples
Exponential functions like y = 6 × 2 x are used to model various real-world phenomena, such as population growth, compound interest, and radioactive decay. For instance, if you invest money in an account that doubles every year, the amount of money you have can be modeled by an exponential function. Evaluating the function at a specific time (e.g., x = 1 year) tells you how much money you'll have at that time. Understanding exponential functions helps in making informed decisions about investments and predicting future outcomes in various scenarios.
To find the value of the function y = 6 ⋅ 2 x at x = 1 , we substitute and calculate to get y = 12 . This simply involves evaluating exponentiation and performing multiplication. Thus, the result is y = 12 .
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