Find the domain of each rational function. h(x)=(x+7)/(x2−49)

Ch. 3 - Polynomial and Rational Functions

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Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

문제 5

4장, 문제 5

Find the domain of each rational function. h(x)=(x+7)/(x²−49)

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Identify the rational function given: \(h(x) = \frac{x+7}{x^{2} - 49}\).

Recall that the domain of a rational function includes all real numbers except where the denominator is zero, because division by zero is undefined.

Set the denominator equal to zero to find the values to exclude: \(x^{2} - 49 = 0\).

Solve the equation \(x^{2} - 49 = 0\) by factoring it as a difference of squares: \((x - 7)(x + 7) = 0\).

Find the roots from the factors: \(x - 7 = 0\) gives \(x = 7\), and \(x + 7 = 0\) gives \(x = -7\). These values are excluded from the domain.

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주요 개념

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

Domain of a Function

The domain of a function is the set of all input values (x-values) for which the function is defined. For rational functions, the domain excludes values that make the denominator zero, as division by zero is undefined.

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding how to simplify and analyze these functions is essential, especially identifying values that cause the denominator Q(x) to be zero.

Factoring Polynomials

Factoring involves rewriting polynomials as products of simpler polynomials. For example, x² - 49 factors as (x - 7)(x + 7). Factoring helps identify zeros of the denominator to determine excluded values from the domain.