Find the domain of each function. f(x)=2xx+1−1f(x) = \sqrt{\frac{2x}{x + 1} - 1} f(x)=x+12x−1

Ch. 3 - Polynomial and Rational Functions

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Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

문제 63

4장, 문제 63

Find the domain of each function. $f(x) = $\sqrt{\frac{2x}{x + 1}$ - 1}$

검증된 단계별 안내

Identify the function given: $f(x) = \sqrt{\frac{2x}{x + 1} - 1}$.

Recall that the expression inside the square root must be greater than or equal to zero for the function to be defined. So, set up the inequality: $\frac{2x}{x + 1} - 1 \geq 0$.

Combine the terms on the left side over a common denominator: $\frac{2x}{x + 1} - \frac{x + 1}{x + 1} \geq 0$, which simplifies to $\frac{2x - (x + 1)}{x + 1} \geq 0$.

Simplify the numerator: $\frac{2x - x - 1}{x + 1} = \frac{x - 1}{x + 1} \geq 0$.

Determine the domain by finding where the rational expression $\frac{x - 1}{x + 1}$ is greater than or equal to zero, and exclude values that make the denominator zero (i.e., $x \neq -1$).

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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) for which the function is defined. For functions involving roots or fractions, certain values may be excluded to avoid undefined expressions like division by zero or taking the square root of a negative number.

Square Root Function Restrictions

When a function includes a square root, the expression inside the root (the radicand) must be greater than or equal to zero. This ensures the output is a real number, as square roots of negative numbers are not defined in the set of real numbers.

Rational Expressions and Denominator Restrictions

For rational expressions, the denominator cannot be zero because division by zero is undefined. Identifying values of x that make the denominator zero is essential to exclude them from the domain.

Find the domain of each function. f(x)=2xx+1−1f(x) = \(\sqrt{\frac{2x}{x + 1}\) - 1} f(x)=x+12x−1

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Domain of a Function

Square Root Function Restrictions

Rational Expressions and Denominator Restrictions

Find the domain of each function. $f(x) = \(\sqrt{\frac{2x}{x + 1}\) - 1}$