The graph of the rational function \( f(x)=\frac{x-2}{x^{2}+4 x-5} \) will have a vertical asymptote at a.) \( x=1 \) and \( x=-5 \) b.) \( x=1 \) and \( x=2 \) c.) \( x=-5 \) and \( x=2 \) d.) \( x=-5, x=1 \), and \( x=2 \)

To determine the vertical asymptotes of the function \( f(x)=\frac{x-2}{x^{2}+4x-5} \), we need to find the values of \( x \) that make the denominator equal to zero, as these are where the function can become undefined. First, we need to factor the quadratic in the denominator: \( x^2 + 4x - 5 = (x + 5)(x - 1) \). Setting \( (x + 5)(x - 1) = 0 \) gives us the points \( x = -5 \) and \( x = 1 \). Thus, the function has vertical asymptotes at \( x = -5 \) and \( x = 1 \). So, the correct answer is: a.) \( x=1 \) and \( x=-5 \) Historical Background: Rational functions have been a fundamental part of mathematical studies since the early days of algebra, dating back to ancient civilizations. These functions often arise in real-world scenarios, such as in physics when dealing with rates and involving proportions that vary inversely. They play a critical role in calculus, particularly when discussing limits and asymptotic behaviors. Real-World Application: Rational functions present in various fields, such as economics and physics. For instance, in economics, the function can model supply and demand relationships, revealing how quantities change based on price fluctuations. Meanwhile, in physics, rational functions might be used to describe material properties under stress or fluid dynamics, helping engineers and scientists predict system behavior in design and analysis.