Skip to main content
Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 63
Chapter 4, Problem 63

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

Verified step by step guidance
1
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., \(x \neq -1\)).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
Recommended video:
3:51
Domain Restrictions of Composed Functions

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.
Recommended video:
02:20
Imaginary Roots with the Square Root Property

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.
Recommended video:
Guided course
02:58
Rationalizing Denominators
Related Practice
Textbook Question

Follow the seven steps to graph each rational function. f(x)=−x/(x+1)

37
views
Textbook Question

Among all pairs of numbers whose difference is 24, find a pair whose product is as small as possible. What is the minimum product?

845
views
Textbook Question

Among all pairs of numbers whose sum is 16, find a pair whose product is as large as possible. What is the maximum product?

1351
views
Textbook Question

In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. g(x) = (4x^2 - 16x + 16)/(2x - 3)

793
views
Textbook Question

Follow the seven steps to graph each rational function. f(x)=2x2/(x2−1)

32
views
Textbook Question

Find the domain of each function. f(x)=14x2−9x+2f(x) = \(\frac{1}{\sqrt{4x^2 - 9x + 2}\)}

521
views