Find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. f(x)=(x²−9)/(x−3)

검증된 단계별 안내

Start by identifying the rational function given: \(f(x) = \frac{x^{2} - 9}{x - 3}\).

Factor the numerator \(x^{2} - 9\) using the difference of squares formula: \(a^{2} - b^{2} = (a - b)(a + b)\), so \(x^{2} - 9 = (x - 3)(x + 3)\).

Rewrite the function using the factored form: \(f(x) = \frac{(x - 3)(x + 3)}{x - 3}\).

Look for common factors in the numerator and denominator. Since \((x - 3)\) appears in both, it can be canceled out, but note that \(x \neq 3\) because it would make the denominator zero.

Determine the vertical asymptotes and holes: the value \(x = 3\) causes the denominator to be zero. Since \((x - 3)\) cancels, there is a hole at \(x = 3\). There are no other values making the denominator zero, so there are no vertical asymptotes.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding the behavior of rational functions involves analyzing their domains, zeros, and discontinuities, which occur where the denominator equals zero.

Vertical Asymptotes

Vertical asymptotes occur at values of x where the denominator of a rational function is zero and the numerator is nonzero, causing the function to approach infinity or negative infinity. They represent lines the graph approaches but never touches.

Holes in the Graph

Holes occur when a factor cancels out from both numerator and denominator, creating a removable discontinuity. At these x-values, the function is undefined, but the limit exists, resulting in a 'hole' in the graph rather than an asymptote.

Find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. f(x)=(x2−9)/(x−3)