Question 2 Find all intercept(s) of the function: \( R(x)=\frac{2 x-3}{5 x^{2}+2 x-3} \). Question 3 Find all vertical asymptote(s) of the function \( q(x)=\frac{2 x-3}{5 x^{2}+2 x-3} \).
Solución de inteligencia artificial de Upstudy
Responder
Solución
Respondido por UpStudy AI y revisado por un tutor profesional
Explicar 
Simplifique esta solución The Deep Dive
To find the intercepts of the function \( R(x) = \frac{2x - 3}{5x^{2} + 2x - 3} \), you should set the numerator equal to zero. This gives you the \( x \)-intercept \( x = \frac{3}{2} \). To find the \( y \)-intercept, substitute \( x = 0 \) into the function, resulting in \( R(0) = -\frac{3}{3} = -1 \). Thus, the intercepts are at \( \left(\frac{3}{2}, 0\right) \) and \( (0, -1) \). To find the vertical asymptotes of \( q(x) = \frac{2x - 3}{5x^{2} + 2x - 3} \), you need to determine when the denominator is zero. Setting \( 5x^{2} + 2x - 3 = 0 \) and solving this quadratic using the quadratic formula gives \( x = \frac{-2 - \sqrt{64}}{10} \) and \( x = \frac{-2 + \sqrt{64}}{10} \). This results in vertical asymptotes at \( x = -1 \) and \( x = \frac{3}{5} \). Want to know something fascinating? Understanding intercepts and asymptotes is crucial not only for mathematics but also for fields like engineering. For example, structural engineers need to calculate loads at specific points, which can be visualized similarly to finding intercepts on a curve. Imagine designing a bridge where every load point needs precision, just like plotting those intercepts! If you’re curious about more sophisticated techniques in calculus, check out L'Hôpital's Rule, which can elegantly handle limits involving indeterminate forms, especially useful when approaching vertical asymptotes. Dive deeper into calculus textbooks or online resources, and you’ll discover techniques that open doors to understanding complex functions and their behaviors. Happy calculating!
