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Chapter 3: Problem 64

In Exercises \(61-64,\) find the domain of each function. $$ f(x)-\sqrt{\frac{x}{2 x-1}-1} $$

Short Answer

Expert verified
The domain of the function is \(-\infty < x \leq 1\) and \(x \neq 1/2\)

Step by step solution

01

Identify the inequality for the square root

Let's set the inside of the square root greater than or equal zero, due to the square root rule.\n\n\(\frac{x}{2x - 1} - 1 \geq 0\)
02

Solve the inequality

Now, solve this inequality.\n\nFirst of all, solve for denominator that cannot be equal to zero: \(2x-1\neq 0\) -> \(x\neq 1/2\).\n\nThen by adding 1 to both sides we get \(\frac{x}{2x - 1} \geq 1\). \n\nCross multiply to remove the denominator: \(x \geq 2x - 1\). \n\nThis simplifies to \(x \leq 1\).
03

Conclusion for the domain of the function.

Combine the results of the inequality with the limitations of the denominator. We find \(x\) must be less than or equal to \(1\) but \(x\) cannot be \(1/2\). So, the domain of the function is all \(x\) such that \(-\infty < x \leq 1\) and \(x \neq 1/2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inequalities
An inequality is a mathematical expression that shows the relationship of one value being greater than, less than, or equal to another. They are pivotal when solving for a domain, especially in functions dealing with square roots and fractions.
For example, consider the expression \( \frac{x}{2x - 1} - 1 \geq 0 \). Inequalities like this help define the range of permissible values for \( x \) in a function.
  • Greater Than or Equal To: This inequality tells us the expression inside a square root must be non-negative (鈮� 0) because square roots of negative numbers are not real numbers.
  • Solving Inequalities: To solve the inequality, we first add or subtract terms to isolate variables, then use operations like multiplication or division, keeping in mind flipping the inequality if multiplying or dividing by a negative.

Understanding inequalities is crucial when working with functions having constraints, like the square root or a denominator, to determine valid input values.
Square Root Function
Square root functions are functions that involve extracting square roots. The primary rule for these functions is the value under the square root must be zero or greater.
For the expression \( \sqrt{\frac{x}{2x-1} - 1} \), the portion inside the square root must satisfy \( \frac{x}{2x-1} - 1 \geq 0 \).
  • Non-Negative Requirement: The value inside the square root, known as the radicand, must not be negative to ensure the square root is real and defined.
  • Checking the Domain: By setting the radicand \( \geq 0 \) and solving, we establish the domain where the function is valid, determining possible \( x \) values.

Square root functions therefore require checking conditions under the radicand to maintain real-number results. This makes them an essential consideration in domain determination.
Rational Expressions
Rational expressions are fractions where both the numerator and the denominator are polynomials. These expressions have specific domain constraints tied to their form.
For instance, in \( \frac{x}{2x-1} \), the denominator cannot be zero because division by zero is undefined.
  • Undefined at Certain Points: To find where a rational function is undefined, set the denominator equal to zero and solve for \( x \). Here, \( 2x-1 eq 0 \) results in \( x eq \frac{1}{2} \).
  • Solving Rational Inequalities: Factors of the denominator or numerator impact the inequality sign, guiding how inequalities are solved and at what points expressions may change sign.

When dealing with rational expressions, identifying values that make the denominator zero is key to ensuring expression validity, helping define the permissible domain of a function.

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The graph of a rational function can never cross a vertical asymptote.

Use a graphing utility to graph \(y-\frac{1}{x^{2}}, y-\frac{1}{x^{4}},\) and \(y-\frac{1}{x^{6}}\) in the same viewing rectangle. For even values of \(n,\) how does changing \(n\) affect the graph of \(y-\frac{1}{x^{n}} ?\)

Solve each inequality in Exercises \(86-91\) using a graphing utility. $$ \frac{x+2}{x-3} \leq 2 $$

In Exercises \(98-101\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The inequality \(\frac{x-2}{x+3}<2\) can be solved by multiplying both sides by \(x+3\), resulting in the equivalent inequality \(x-2<2(x+3)\)

Use the position function $$ s(t)--16 t^{2}+v_{0} t+s_{0} $$ \(\left(v_{11}=\text { initial velocity, } s_{0}-\text { initial position, } t-\text { time }\right)\) to answer Exercises \(75-76\) You throw a ball straight up from a rooftop 160 feet high with an initial velocity of 48 feet per second. During which time period will the ball's height exceed that of the rooftop?
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