Problem 64 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The domain of the function \(f(x)=\sqrt{\frac{x}{2 x-1}-1}\) is: \((x: x \neq 1/2) \cap (x: x \geq (3 +/- sqrt(5)) / 2)\)

Step 1: Find restriction for denominator

First, set the denominator of the fraction to not be equal to zero: \(2x - 1 \neq 0\), which translates to: \(x \neq 1/2\).

Step 2: Set expression under square root to be greater than or equal to 0

Next, to avoid getting an imaginary number when taking the square root, set the entire expression under the square root to be greater than or equal to 0: \(\frac{x}{2x-1}-1 \geq 0\). Solving this inequality will provide the initial possible domain of the function.

Step 3: Solve the inequality

Solving for the inequality we get: \(x \geq (3 +/- sqrt(5)) / 2\). This means that x can either be equal to or larger than both of these values.

Step 4: Combine the restrictions

Due to step 1 and 3, the overall domain of the function would be given by the intersection of the restrictions. Thus, the domain of this function is \((x: x \neq 1/2) \cap (x: x \geq (3 +/- sqrt(5)) / 2)\). It is important to note that these restrictions must be satisfied simultaneously for any value of x.

Key Concepts

Understanding Inequalities in Function Domains

Inequalities help us determine the set of values (known as the domain) that a function can potentially accept without resulting in mathematical inconsistencies.
In the context of functions involving fractions and square roots, inequalities are especially important.
They prevent us from dividing by zero or taking the square root of negative numbers. Let's see how inequalities are used in this example to find the domain of the function \( f(x)=\sqrt{\frac{x}{2x-1}-1} \).

• First, recognize that the expression under the square root, \( \frac{x}{2x-1}-1 \), needs to be greater than or equal to 0. Otherwise, the result would be an imaginary number.
• Thus, we set the inequality \( \frac{x}{2x-1} - 1 \geq 0 \) and solve it to understand which values x can take.

By solving this inequality, we not only avoid imaginary numbers but ensure every input keeps the function within real numbers, defining our possible domain.

Square Root Restrictions and Their Impact

Square root restrictions are critical since the square root of a negative number is not defined within the realm of real numbers.
Therefore, any function involving a square root must consider what values might turn the expression under the root negative.In our function \( f(x)=\sqrt{\frac{x}{2x-1}-1} \), the presence of a square root means:

• The expression \( \frac{x}{2x-1}-1 \) must be equal to or greater than zero.
• This ensures the entire value under the square root remains non-negative.

By constraining x through these restrictions, we define the subset of numbers that keep the function input valid.
This subset forms part of the domain of the function.

Rational Functions and Their Considerations

Rational functions, such as \( \frac{x}{2x-1} \), are ratios of polynomials.
Their domains are typically affected by their denominators, as division by zero is undefined.
Hence, you must set restrictions for these values.To prevent division by zero:

• We determine where the denominator, in this case \(2x-1\), equals zero, and exclude these x-values from the domain.
• Here, setting \(2x-1 eq 0\) leads us to find that \(x eq \frac{1}{2}\).

It's crucial to combine these restrictions with others (like those from square roots) to fully define the function's domain.
Each restriction works together to pinpoint safely usable x-values.