Problem 9 - Differential Equations and Linear Algebra

Question

Determine \(f(t-a)\) for the given function \(f\) and the given constant \(a\). $$f(t)=\frac{t}{t^{2}+4}, \quad a=1$$.

Step-by-Step Solution

Answer

The short answer for the given function \(f(t)=\frac{t}{t^2+4}\) and constant \(a=1\) is: \(f(t-1)=\frac{t-1}{t^2 - 2t + 5}\).

Step 1: Replace t with (t-1) in the original function

Substituting \(t\) with \((t-1)\) in \(f(t)=\frac{t}{t^2+4}\), we get: \(f(t-1)=\frac{t-1}{(t-1)^2+4}\)

Step 2: Simplify the expression

We need to simplify the denominator by expanding \((t-1)^2\) and combining the result with 4: \((t-1)^2 = (t-1)(t-1) = t^2 - 2t +1\) Now replace \((t-1)^2\) in the denominator with \(t^2 - 2t +1\): \(f(t-1)=\frac{t-1}{t^2 - 2t +1 + 4}\) Simplify further: \(f(t-1)=\frac{t-1}{t^2 - 2t + 5}\)

Step 3: Write the final result

Now we have found the function \(f(t-1)\) for the given function \(f(t)\) and the given constant \(a=1\). The resulting function is: \(f(t-1)=\frac{t-1}{t^2 - 2t + 5}\)

Key Concepts

Function Transformation

Function transformation is a crucial concept in mathematics, especially when dealing with differential equations. It's akin to tweaking a function in various ways to make it fit a specific need or analysis. Here's a simple take on what function transformation entails.

A function can be transformed in various ways, such as by scaling, shifting, or even reflecting it. In our exercise, we focus on the transformation known as a translation. This involves moving the function left or right along the horizontal axis.

For the given function transformation, we replace every instance of the variable \(t\) in the function \(f(t)\) with \(t-a\) (here \(a = 1\)). This type of transformation is particularly useful in aligning functions or graphs for easier comparison or integration. Remember, the main goal of transforming a function is to explore how it behaves under different conditions and shifts.

Shift Operator

The shift operator is a simple yet powerful tool in mathematics. It involves shifting the entire graph of a function along the axis. In our given function \(f(t)=\frac{t}{t^{2}+4}\), we applied a shift operator to modify the original function.

The shift operator keeps the structure of the function intact while changing its position along the horizontal axis. In the problem, the shift operator is applied by replacing \(t\) with \(t-1\). This moves the entire graph of the function one unit to the right. Think of it as giving the function a small nudge along the \t\-axis.

• Benefits: It helps in understanding variations in function calues as \(t\) changes.
• Importance: Critical in scenarios like solving differential equations where assessing shifts can simplify the problem.

Using shift operators can dramatically simplify complex expressions in a differential equation, making it easier to understand and solve the functions.

Function Simplification

Function simplification involves reducing a function into its most simple form. The goal is to transform an expression into its simplest equivalent, making it easier to work with. In our exercise, we have simplified the function \(f(t-1)=\frac{t-1}{(t-1)^2+4}\).

To simplify this function, we focus on carefully expanding \( (t-1)^2 \) in the denominator. The expansion results in \(t^2 - 2t + 1\). Adding up with the constant \(4\) gives us the simplified form as \(t^2 - 2t + 5\). This helps in easily analyzing the behavior of the function without complicated terms.

Function simplification is essential in mathematics for several reasons:

• Reduces computational complexity.
• Makes functions more readable.
• Facilitates understanding and applying mathematical operations like integration or differentiation.

By simplifying functions, you make it significantly easier for mathematical software and tools to process and evaluate them, thereby enhancing computational efficiency.