Question 2 Find all intercept(s) of the function: \( R(x)=\frac{2 x-3}{5 x^{2}+2 x-3} \)

To find the intercepts of the function \( R(x) = \frac{2x - 3}{5x^2 + 2x - 3} \), we start with the x-intercepts. Set the numerator equal to zero: \( 2x - 3 = 0 \). Solving this gives \( x = \frac{3}{2} \). Therefore, the x-intercept is \( \left(\frac{3}{2}, 0\right) \). For the y-intercept, set \( x = 0 \). Plugging this into the function, we get \( R(0) = \frac{2(0) - 3}{5(0)^2 + 2(0) - 3} = \frac{-3}{-3} = 1 \). So, the y-intercept is \( (0, 1) \). These two points, \( \left(\frac{3}{2}, 0\right) \) and \( (0, 1) \), are the intercepts! --- Did you know that the concept of intercepts is not only useful in algebra but also in real-world scenarios like economics and engineering? For instance, in a supply and demand graph, the x-intercept may represent the maximum quantity of goods, while the y-intercept could indicate the highest price consumers are willing to pay. When working with rational functions like this one, a common mistake is to overlook vertical intercepts arising from the denominator. Always check for restrictions that may make the denominator equal to zero, as this often indicates asymptotes rather than valid intercepts!