91Ó°ÊÓ

When evaluating a function, we aim to find the output value based on given inputs. Imagine you have a machine, where you input certain values, and it calculates something magical. The process of evaluation involves following this machine's rules to see what comes out.For the function \( f(x, t) = \mathrm{e}^{2x} \), you are provided with a blueprint of how the machine works. Here, \( x \) is the crucial input, while \( t \) doesn't affect the function calculation in this exercise.So, when you see \( f(0.5, 3) \), it’s asking "What happens when \( x \) is 0.5?" The value of \( t \) doesn't change the outcome. That's why once you input \( x = 0.5 \), you follow the function’s instructions to evaluate it.

Substitution in functions means replacing variables in a function with specific numbers. Think of it like swapping a placeholder in a recipe with the actual ingredient you're using. For \( f(0.5, 3) = \mathrm{e}^{2(0.5)} \), you're instructed to swap out \( x \) with 0.5. Remember, it’s like filling in a blank spot on a form with given information.The function \( f(x, t) = \mathrm{e}^{2x} \) is a handy rule that tells you what to do with \( x \). Substituting turns abstract symbols into concrete numbers, simplifying the process down to numbers we can easily handle.

• Identify which values to replace in the function rule.
• Put the numbers into the function in place of the variables.

Once substituted, the expression transforms, preparing it for simplification.

Simplifying expressions is like tidying up your workspace — it makes everything clear and manageable. When an expression is simplified, it’s reduced to its simplest form, eliminating confusion and excessive detail.From \( \mathrm{e}^{2(0.5)} \), we simplify to \( \mathrm{e}^{1} \). This step involves simple arithmetic: multiplying the numbers in the exponent \( 2 \times 0.5 = 1 \). Why simplify?

• It makes understanding easier.
• It prepares the expression for final evaluation.

In this case, once you've simplified, you can directly evaluate the expression, recognizing \( \mathrm{e}^{1} \) as \( e \), the mathematical constant. This approach helps to view calculations more clearly and ensure accuracy before finalizing results.