Find the domain of each function. f(x)=x/(x2+4x−21) f(x) = $$x/(x^2 + 4x -21)$$

Ch. 2 - Functions and Graphs

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Blitzer 8th Edition

Ch. 2 - Functions and Graphs

문제 79

3장, 문제 79

Find the domain of each function. $f (x) = x / (x^{2} + 4 x - 21)$

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Identify the function given: $f(x) = \frac{x}{x^2 + 4x - 21}$.

Recall that the domain of a 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: $x^2 + 4x - 21 = 0$.

Factor the quadratic equation: find two numbers that multiply to $-21$ and add to $4$. This gives $(x + 7)(x - 3) = 0$.

Solve each factor for zero: $x + 7 = 0$ gives $x = -7$, and $x - 3 = 0$ gives $x = 3$. 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 possible 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.

Factoring Quadratic Expressions

Factoring involves rewriting a quadratic expression as a product of two binomials. This helps identify the roots or zeros of the quadratic, which are critical for determining values that make the denominator zero in rational functions.

Solving Quadratic Equations

Solving quadratic equations means finding the values of x that satisfy the equation when set equal to zero. These solutions indicate points where the denominator of a rational function is zero and must be excluded from the domain.

Find the domain of each function. f(x)=x/(x2+4x−21) f(x) = x/(x^2 + 4x -21)