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)=x/(x+4)

검증된 단계별 안내

Identify the rational function given: \(f(x) = \frac{x}{x+4}\).

Determine the values of \(x\) that make the denominator zero, since these are potential vertical asymptotes or holes. Set the denominator equal to zero: \(x + 4 = 0\).

Solve for \(x\): \(x = -4\). This is the value where the function is undefined.

Check if the factor \(x + 4\) cancels with any factor in the numerator. Since the numerator is \(x\), which does not contain \(x + 4\) as a factor, there is no cancellation.

Conclude that \(x = -4\) is a vertical asymptote, and there are no holes in the graph of \(f(x)\).

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주요 개념

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

Vertical Asymptotes

Vertical asymptotes occur where a rational function's denominator is zero and the numerator is nonzero, causing the function to approach infinity or negative infinity. To find them, set the denominator equal to zero and solve for x, excluding any values that also make the numerator zero.

Holes in the Graph

Holes occur in the graph of a rational function when a factor cancels out from both the numerator and denominator, resulting in an undefined point rather than an asymptote. To identify holes, factor both numerator and denominator, cancel common factors, and find the x-values that make these factors zero.

Rational Functions and Domain

A rational function is a ratio of two polynomials. Its domain includes all real numbers except where the denominator is zero, as division by zero is undefined. Understanding the domain helps in identifying points of discontinuity such as vertical asymptotes and holes.