What are the vertical asymptotes of [tex]f(x)=\frac{10}{x^2-1}[/tex]?
Answer (2)
Find where the denominator of f ( x ) = x 2 − 1 10 equals zero by solving x 2 − 1 = 0 .
Factor the denominator to get ( x − 1 ) ( x + 1 ) = 0 .
Solve for x , yielding x = 1 and x = − 1 .
The vertical asymptotes are x = − 1 and x = 1 , since the numerator is non-zero at these points. x = − 1 , x = 1
Explanation
Understanding Vertical Asymptotes We are given the function f ( x ) = x 2 − 1 10 and asked to find its vertical asymptotes. Vertical asymptotes occur at values of x where the denominator of the function is equal to zero and the numerator is non-zero.
Setting the Denominator to Zero To find the vertical asymptotes, we need to solve the equation x 2 − 1 = 0 .
Factoring the Quadratic We can factor the quadratic expression as ( x − 1 ) ( x + 1 ) = 0 .
Finding Potential Asymptotes This gives us two possible solutions: x = 1 and x = − 1 .
Verifying the Numerator Now we need to check if the numerator is non-zero at these values. The numerator is 10, which is non-zero at both x = 1 and x = − 1 . Therefore, the vertical asymptotes are x = 1 and x = − 1 .
Final Answer The vertical asymptotes of the function f ( x ) = x 2 − 1 10 are x = 1 and x = − 1 .
Examples
Understanding vertical asymptotes is crucial in various fields. For instance, in physics, when analyzing the behavior of electric fields around point charges, the field strength approaches infinity as you get closer to the charge, creating a vertical asymptote on a graph of field strength versus distance. Similarly, in economics, cost functions may have vertical asymptotes representing capacity limits, where costs skyrocket as production nears maximum capacity. Recognizing and interpreting these asymptotes helps in making informed decisions and avoiding critical operational limits.
The vertical asymptotes of the function f ( x ) = x 2 − 1 10 are at x = 1 and x = − 1 . These occur where the denominator equals zero while the numerator remains non-zero. Specifically, the function approaches infinity at these values.
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