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In every function, you deal with two types of values: inputs and outputs. An input value is what you use to calculate something in the function, often represented by a variable like \( t \) or \( x \). Think of it as the starting point; what you feed into the function. In our example, \( A(t)=32 e^{0.5 t} \), the input value is \( t \), which tells us the time variable's value we're considering is \( t = 15 \).

Outputs, on the other hand, are the results of the function after processing the input. They show us what the output variable becomes given a specific input. In our exercise, \( A(t) \) represents this output value. After computing, this gives us the corresponding value of the function when \( t = 15 \). Identifying these correctly ensures you are plugging the right values into the function to receive accurate results.

Evaluating a function is like following a recipe. Once you have the input, you substitute it into the function to find what outcome or result it yields. Let's break down the given function evaluation in our example. The function is \( A(t) = 32 e^{0.5 t} \). Your task is to substitute \( t = 15 \) into the function and find \( A(15) \).

First off, replace \( t \) with 15, giving you \( A(15) = 32 e^{0.5 imes 15} \). Next, simplify inside the exponential: calculate \( 0.5 imes 15 \), which equals 7.5. Now, our function reads \( A(15) = 32 e^{7.5} \).

Depending on your calculator settings, you can find \( e^{7.5} \). Suppose you find this to be roughly 1808.042. Then, multiply this result by the coefficient 32 to get your answer \( A(15) \approx 57857.344 \). Understanding function evaluation is critical as it ensures you are correctly working through the operations in sequence.

Rounding numbers is an essential mathematical skill, especially in exercises where precise and cleaner results are necessary. Often, we're required to round to a specific number of decimal places, which helps declutter numbers and enhances readability.

In the given exercise, after evaluating \( A(15) = 32 \times 1808.042 \), you end up calculating \( 57857.344 \). Here, you round the result to three decimal places, meaning the digits after the decimal point are precise up to three digits.

To round correctly, look at the fourth digit after the decimal point. If it is 5 or more, increase the third digit by one. If it's less, leave the third digit as it is. That's how \( 57857.344 \) remains \( 57857.344 \) when rounded to three decimal places, as the number is already in the desired format. Rounding isn’t just a mathematical precision step; it's a way to handle real-world data efficiently, ensuring that information is conveyed in a precise yet simplified way.