Problem 44 - Algebra 1

Question

Find the domain of the function. $$y=\frac{\sqrt{4-x}}{x}$$

Step-by-Step Solution

Answer

The domain of the function \(y=\frac{\sqrt{4-x}}{x}\) is \(x \in (-\infty, 0) \cup (0,4]\)

Step 1: Determine when the denominator equals zero

Equation \(x=0\) will be solved as the denominator cannot be zero.

Step 2: Solve for values where the number under the square root is negative

The inequality \(4-x \geq 0\) must be solved because taking a square root of a negative number is undefined in real numbers.

Step 3: Combine these two conditions

Combine these two conditions for the domain. The first condition gave that \(x \neq 0\) and the second condition gave that \(x \leq 4\), so the domain of the function is \(x \in (-\infty, 0) \cup (0,4]\).

Key Concepts

Square Root Function

The square root function is a common mathematical function characterized by the radical symbol "√." It is important to understand the properties and constraints when dealing with square roots, particularly in relation to the domain, or set of possible input values.

For a square root function of the form \( \sqrt{4-x} \), the expression inside the square root (known as the radicand) must be non-negative to yield a real number. This is because the square root of a negative number is not defined in the set of real numbers.

To find the valid input values for the square root function, we solve the inequality:

• \( 4-x \geq 0 \)

Rearranging gives us \( x \leq 4 \), meaning any input value for \( x \) must be less than or equal to 4. This restriction ensures that the square root is always taking the root of a non-negative number, staying within the real number system. In this particular function, the square root is the numerator of a fraction, meaning this constraint must be satisfied for the fraction to be defined.

Rational Function

Rational functions are comprised of a numerator and a denominator, each of which can be any polynomial expression. The domain of a rational function is all real numbers except for those which make the denominator zero, because division by zero is undefined.

In our specific function \( y=\frac{\sqrt{4-x}}{x} \), the denominator is simply \( x \). Therefore, we must exclude \( x = 0 \) from the domain.

• This is determined by setting the denominator equal to zero and solving for \( x \):
• \( x = 0 \)

Since dividing by zero is undefined, \( x = 0 \) is not in the domain of the function.

This means the domain consists of all real numbers except zero, combined with the constraint from the square root function.

Inequalities

Inequalities in mathematics involve expressions where values are not necessarily equal, but instead greater than or less than each other. Understanding inequalities is crucial in analyzing functions like the one given in this exercise, as they help define the domain.

For the current function, we have two critical inequalities to consider:

• \( 4-x \geq 0 \)
• \( x eq 0 \)

The solution to the first inequality reveals that \( x \leq 4 \) ensures the radicand is non-negative, while the second inequality eliminates \( x = 0 \) to avoid division by zero.

Combining these inequalities, the domain is identified as all \( x \) such that \(-\infty < x \leq 4\), excluding \( x = 0 \). Therefore, the domain is expressed as \( x \in (-\infty, 0) \cup (0,4] \). This combined result shows all values that make the function well-defined, ensuring both conditions are met simultaneously.