Problem 65 - College Algebra

Question

Find the domain of each function. $$ f(x)=\frac{1}{\sqrt{x-3}} $$

Step-by-Step Solution

Answer

The domain of the function \(f(x) = \frac{1}{\sqrt{x-3}}\) is \(x > 3\).

Step 1: Identify the Original Function

Identify that the given function \(f(x)=\frac{1}{\sqrt{x-3}}\) is a combination of a reciprocal and a square root function.

Step 2: Set the Function Inside the Radical Greater than Zero

Set the inside function (the one inside the square root) which is \(x-3\) greater than zero. The reason we do this is because we cannot take the square root of a negative number within the real number system, thus for the square root function to be defined, \(x-3\) must be more than or equal to zero. So, solve the inequality \(x-3 > 0\).

Step 3: Solve the Inequality

Solving the inequality \(x-3 > 0\), we will get \(x > 3\). However, we have to be careful because 3 is not included in the solution because substituting 3 results in division by zero which is undefined.

Step 4: State the Domain

The solution to the inequality give us the domain of the function. So, the domain of the function \(f(x)=\frac{1}{\sqrt{x-3}}\) is \(x > 3\). That means the function is defined for all real numbers greater than 3.

Key Concepts

Reciprocal Function

A reciprocal function is a function that involves the division of 1 by a variable. In mathematical terms, it is often expressed as \( y = \frac{1}{x} \). This basic form tells us that as \( x \) approaches zero, the function's value grows infinitely positive or negative, leading to a graph where \( x = 0 \) is a vertical asymptote.
These characteristics make the reciprocal function interesting:

• The function is undefined at \( x = 0 \), resulting in a vertical asymptote.
• When positive numbers are used, the function yields positive results.
• Negative inputs result in negative outputs, reflecting symmetry with respect to the origin.

In our problem, the reciprocal function is combined with another function. This implies that certain domain limitations may arise—here, due to dividing by the square root.

Square Root Function

The square root function is defined as \( y = \sqrt{x} \). This function maps a number to its square root, and it is only defined for non-negative inputs within the real number system.

• The reason behind this restriction is that square roots of negative numbers are not real.
• The graph of a simple square root function is a curve that starts at the origin (0,0) and rises slowly to the right.
• Any output is a non-negative real number.

In our specific exercise, \( \sqrt{x-3} \) means \( x-3 \) must be greater than zero to have real number outputs. This consideration is critical to determining the entire domain of the function.

Inequality Solving

In the context of finding the function's domain, solving inequalities helps identify where the function is defined.
For instance, in our exercise, we have \( x-3 > 0 \) to ensure the function is inside the square root is defined in the realm of real numbers.

• Simplifying the inequality \( x-3 > 0 \) involves isolating \( x \).
• This translates to \( x > 3 \), which means only numbers greater than 3 can be inputs to the function.

Note the importance of strict inequality signs here because equality (\( x = 3 \)) would result in division by zero. The understanding of such inequalities ensures the function remains defined and unrestricted by non-real values.

Real Numbers

Real numbers encompass all numbers along the continuous number line, including both rational and irrational numbers.

• Rational numbers can be expressed as fractions, like \( \frac{1}{2} \), while irrational numbers cannot, such as \( \sqrt{2} \).
• The domain of most functions is limited to real numbers, for predictability and defined results.
• When excluding certain values (like in our exercise), those values are often points where functions become undefined or imaginary.

Understanding real numbers helps in defining the domain correctly, ensuring that inputs to functions are valid for true, real-world calculations without falling into the realm of imaginary or complex numbers.