Question

If \(f(x, t)=\mathrm{e}^{2 x}\) find \(f(0.5,3)\).

Short answer

Answer: The value of the function at x = 0.5 and t = 3 is e.

Step-by-step solution

Identify the function

The given function is: \(f(x, t) = \mathrm{e}^{2x}\)

Plug in the values

Insert the values x = 0.5 and t = 3 into the function: \(f(0.5, 3) = \mathrm{e}^{2(0.5)}\)

Simplify and solve

Simplify the exponent: \( \mathrm{e}^{2(0.5)} = \mathrm{e}^{1}\) Now, evaluate the exponential function with an exponent of 1: \(f(0.5, 3) = \mathrm{e}^{1} = e\) So, the function evaluated at \(f(0.5, 3)\) equals e.

Key concepts

Evaluation of Functions

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

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.

Simplification of Expressions

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.