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Function substitution is a powerful technique used in mathematics to transform functions by replacing variables within them. In the context of Laplace transforms, function substitution plays a crucial role as it allows one to explore how changes in the input of a function alter its output.
In our exercise, we start with the function \( F(s) = \frac{1}{s+1} \). The task involves finding what happens when we substitute \( s \) with \( 5s \) to find \( F(5s) \). By replacing \( s \) with \( 5s \) within the function, we change its expression, specifically altering its domain and, subsequently, its behavior.
In practice, substituting \( s \) with \( 5s \) gives us a new formula:

• Original function: \( \frac{1}{s+1} \)
• Substitute \( s \) with \( 5s \): \( \frac{1}{5s+1} \)

This substitution reflects how the function behaves when the input is scaled by a factor of 5, effectively shrinking its output for each unit change in \( s \). Function substitution is an invaluable tool in calculus and transforms, providing a different perspective on how functions represent transformations.

A rational function is a type of function that can be expressed as the ratio of two polynomials. These are of particular interest due to their wide-ranging applications in both practical and theoretical contexts.
The specific function in our exercise, \( \frac{1}{s+1} \), is a rational function because it is the quotient of the polynomial 1, which is constant, over the polynomial \( s+1 \). Rational functions often possess characteristics such as vertical asymptotes, which occur where the denominator equals zero, and horizontal asymptotes, determined by the growth of the numerator and denominator as \( s \) approaches infinity.
By interacting with the denominator through function substitution, where we introduce \( 5s \), we modify the function's behavior and its graphical representation while retaining its nature as a rational function:
Learning about rational functions helps students understand behaviors like asymptotes and intercepts, crucial when analyzing how functions behave over their domains.

Multiplication of functions involves creating a product of one function with a constant or another function. In our specific problem, the multiplication step involves multiplying the substituted function by a factor of 5.
After substituting \( s \) with \( 5s \), the resulting function is \( F(5s) = \frac{1}{5s+1} \). The next step involves multiplying this function by 5, as directed by the problem's requirement to find \( 5F(5s) \).
Here’s how it's done:

• Function after substitution: \( \frac{1}{5s+1} \)
• Multiply by 5: \( 5 \times \frac{1}{5s+1} \)
• This results in: \( \frac{5}{5s+1} \)

This simple multiplication changes the scalar of the output, affecting the amplitude but not stretching the function's domain. Such transformations are often used in adjusting signal magnitudes in systems that use Laplace transforms, showcasing a real-world application of function multiplication.