Determine whether the following statements are true and give an explanation or counterexample.
The line x=−1 is a vertical asymptote of the function f(x) =x^2 − 7x + 6 / x^2 − 1.

검증된 단계별 안내

Step 1: Identify the function f(x) = \(\frac{x^2 - 7x + 6}{x^2 - 1}\). This is a rational function, which means it is a fraction where both the numerator and the denominator are polynomials.

Step 2: Determine the points where the denominator is zero, as these are potential vertical asymptotes. Set the denominator equal to zero: x^2 - 1 = 0.

Step 3: Solve the equation x^2 - 1 = 0. This can be factored as (x - 1)(x + 1) = 0, giving the solutions x = 1 and x = -1.

Step 4: Check if x = -1 is a vertical asymptote by ensuring it is not a removable discontinuity. Factor the numerator: x^2 - 7x + 6 = (x - 1)(x - 6).

Step 5: Since x = -1 does not cancel out with any factor in the numerator, it is indeed a vertical asymptote. Therefore, the statement is true.

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주요 개념

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Vertical Asymptotes

A vertical asymptote occurs in a function when the function approaches infinity or negative infinity as the input approaches a certain value. This typically happens at values of x that make the denominator of a rational function equal to zero, provided that the numerator does not also equal zero at that point.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. The general form is f(x) = P(x)/Q(x), where P and Q are polynomials. Understanding the behavior of rational functions, especially their asymptotic behavior, is crucial for analyzing their graphs and determining points of discontinuity.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors, which can simplify the analysis of functions. For example, the function f(x) = (x^2 - 7x + 6)/(x^2 - 1) can be factored to identify points where the function is undefined, helping to determine the presence of vertical asymptotes.

Determine whether the following statements are true and give an explanation or counterexample.The line x=−1 is a vertical asymptote of the function f(x) =x^2 − 7x + 6 / x^2 − 1.