Questions: Write a formula for the polynomial shown below, using the smallest powers possible: f(x)=

Solution Steps

The graph has vertical asymptotes at \( x = -1 \) and \( x = 1 \). This suggests that the function has factors in the denominator that become zero at these points.

The horizontal asymptote is \( y = 1 \). This indicates that as \( x \) approaches \( \pm \infty \), the function approaches 1. This suggests that the degree of the numerator and the denominator are the same, and the leading coefficients' ratio is 1.

Given the vertical asymptotes and the horizontal asymptote, the function can be written in the form: \[ f(x) = \frac{ax^2 + bx + c}{(x+1)(x-1)} \]

Since the horizontal asymptote is \( y = 1 \), the leading coefficients of the numerator and denominator must be equal. Therefore, the numerator should be \( x^2 \) to match the \( x^2 \) in the denominator: \[ f(x) = \frac{x^2}{(x+1)(x-1)} \]

Final Answer