Find the vertical asymptotes of the rational function.
[tex]f(x)=\frac{x(x-1)}{x^3+25 x}[/tex]
A. [tex]x=0, x=-25[/tex]
B. [tex]x=0, x=-5, x=5[/tex]
C. [tex]x=-5, x=5[/tex]
D. none
Answer (1)
Factor the denominator of the rational function: x 3 + 25 x = x ( x 2 + 25 ) .
Find the real roots of the denominator: x = 0 .
Check if the numerator is also zero at x = 0 : 0 ( 0 − 1 ) = 0 . Since both numerator and denominator are zero, there is a hole at x = 0 .
Simplify the function: f ( x ) = x 2 + 25 x − 1 . The denominator x 2 + 25 has no real roots, so there are no vertical asymptotes. none
Explanation
Problem Analysis We are given the rational function f ( x ) = x 3 + 25 x x ( x − 1 ) . We need to find its vertical asymptotes. Vertical asymptotes occur at values of x where the denominator is zero and the numerator is non-zero.
Factoring the Denominator First, let's factor the denominator: x 3 + 25 x = x ( x 2 + 25 ) .
Finding Roots of the Denominator Now, we find the values of x for which the denominator is zero: x ( x 2 + 25 ) = 0 . This gives us x = 0 or x 2 + 25 = 0 . The solutions to x 2 + 25 = 0 are x = ± 5 i , which are imaginary numbers. Since we are looking for vertical asymptotes, we are only interested in real values of x . Thus, x = 0 is the only real solution from the denominator.
Checking the Numerator Next, we check if the numerator is also zero at x = 0 . The numerator is x ( x − 1 ) . When x = 0 , the numerator is 0 ( 0 − 1 ) = 0 . Since both the numerator and denominator are zero at x = 0 , there is a hole at x = 0 , not a vertical asymptote.
Simplifying the Function However, we can simplify the function by canceling the common factor of x from the numerator and denominator: f ( x ) = x ( x 2 + 25 ) x ( x − 1 ) = x 2 + 25 x − 1 , for x = 0 . Now, we look for the values of x where the denominator x 2 + 25 is zero. As we found earlier, the solutions to x 2 + 25 = 0 are x = ± 5 i , which are not real numbers. Therefore, there are no real values of x for which the denominator is zero.
Conclusion Since there are no real values of x that make the denominator zero after simplification, there are no vertical asymptotes for this rational function.
Examples
Consider designing a bridge where the load distribution is modeled by a rational function. Vertical asymptotes would represent points where the load becomes infinitely large, indicating potential structural weaknesses. Identifying and avoiding these points is crucial for ensuring the bridge's stability and safety. Similarly, in electrical circuit design, asymptotes in transfer functions can indicate frequencies at which the circuit becomes unstable or exhibits resonance, which engineers must manage to ensure proper circuit operation.
