Problem 63 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The domain of the function \(f(x)=\sqrt{\frac{2 x}{x+1}-1}\) is \(x \in (-\infty, -1) \cup [1, \infty)\).

Step 1: Identifying values that make denominator zero

Solving the equation \(x+1 = 0\). This gives \(x = -1\). So, \(x\) can be any real number except -1.

Step 2: Identifying values that make radicand non-negative

Solving the equation \(\frac{2x}{x+1}-1 \geq 0\). Multiplying through by \(x+1\) (remembering that this affects the inequality if \(x+1<0\)), you get the inequalities \(2x \geq x + 1\) if \(x+1 > 0\) or \(2x \leq x+1\) if \(x+1 < 0\). Solving these gives \(x \geq 1\) or \(x < -1\). Combining this with Step 1, the valid values of \(x\) are all \(x<-1\) or \(x\geq1\).

Step 3: Combining Results

The final step is to combine the results of steps 1 and 2 to identify the domain of the function. From step 1, any real number except -1 is a reasonable value for the denominator of the expression. From step 2, valid values are those that make the radicand non-negative, \(x<-1\) or \(x\geq1\). Combining these results, the domain of function \(f\) is \(x \in (-\infty, -1) \cup [1, \infty)\). Meaning, all real numbers less than -1 and all real numbers greater than or equal to 1.

Key Concepts

Real Numbers

The term "real numbers" refers to every number that can be found on the number line. This includes both the rational numbers, such as fractions and integers, and irrational numbers, which cannot be expressed as simple fractions. Essentially, real numbers encompass everything from negative numbers, zero, to positive numbers and even decimal points.

When considering the domain of a function, real numbers play a crucial role. This is because the domain is basically the set of input values (or "x" values) that are possible for the function. For most functions, the domain is all real numbers, but sometimes certain values are excluded due to restrictions like division by zero or taking a square root of a negative number.

These limitations highlight why completely understanding real numbers helps in determining the domain of functions.

Inequality Solving

Inequality solving is a method used to find a range of values that satisfy an inequality expression. In mathematics, inequalities are statements that show the relation between two expressions that are not equal. Inequality symbols include ">", "<", "≥", and "≤".

In this function, \ \( \frac{2x}{x+1} - 1 \geq 0 \)
\ we solve the inequality to determine all possible values of \(x\) that make the expression non-negative. This involves rearranging the inequality, finding critical points, and testing intervals between these points.

• First, find where the expression equals zero by solving for \(2x = x + 1\).
• This gives us a potential boundary point, \(x = 1\).
• Next, test intervals around this point to ensure where the inequality holds true.

By solving the inequality, we determine which values satisfy the condition and further refine the domain of the function based on these valid \(x\) values.

Radicand

In functions involving a square root, the expression under the radical sign is called the radicand. To ensure the output is a real number, this value must be non-negative. This is because the square root of a negative number would be an imaginary number, which falls outside the realm of real numbers used in typical function domains.

For the function \ \( f(x) = \sqrt{\frac{2x}{x+1} - 1} \),
the radicand is \ \( \frac{2x}{x+1} - 1 \).
\ This expression needs to be greater than or equal to zero to ensure the square root can be calculated legitimately. By finding when the radicand is non-negative, we determine the \(x\) values viable under the square root, ensuring they are included while determining the domain.

Denominator Zero

A crucial aspect in finding the domain of a rational function concerns the denominator of any fractions in the function. If the denominator becomes zero, it makes the entire fraction undefined because division by zero is not possible. Thus, such values are excluded from the domain.

In the function \ \( f(x) = \sqrt{\frac{2x}{x+1} - 1} \),
\ the denominator is \(x + 1\).

To keep the fraction valid, solve for when this value equals zero:

• Set \(x + 1 = 0\), which solves to \(x = -1\).

This solution means \(x=-1\) makes the denominator zero and thus, must be excluded from the domain of the function. By identifying these values, we ensure the domain only includes valid real numbers, establishing proper limits for the function's input.