Question

Find the domain of each function. \(f(x)=\frac{x}{\sqrt{x-4}}\)

Short answer

The domain is \((4, \infty)\).

Step-by-step solution

Identify the constraints on the denominator

The function is given by \(f(x)=\frac{x}{\sqrt{x-4}}\). The denominator, \(\sqrt{x-4}\), must be defined and not equal to zero. Thus, \( x-4 \) must be positive.

Set up the inequality for the denominator

For the denominator \(\sqrt{x-4}\) to be defined and not zero, we must have: \(x-4 > 0\).

Solve the inequality

Solving \(x-4 > 0\) gives: \(x > 4\).

Key concepts

Rational Functions

Rational functions are fractions where both the numerator and the denominator are polynomials.
In simpler terms, they are just ratios of two polynomials.
The general form of a rational function is \( \frac{P(x)}{Q(x)} \), where both P(x) and Q(x) are polynomials.
For the function to be defined, the denominator Q(x) cannot be zero, because division by zero is undefined.
In this exercise, the function is given by \( f(x) = \frac{x}{\sqrt{x-4}} \). Here's a breakdown of the function elements:

• The numerator: x
• The denominator: \( \sqrt{x-4} \)

Understanding these parts will help us tackle finding the domain.

Inequalities

Inequalities are mathematical expressions that describe the relative size or order of two values.
They tell us if one value is greater than, less than, or equal to another value.
In this problem, we need to work with the inequality \( x-4 > 0 \).
When solving inequalities, we perform similar steps as solving equations.
However, it's important to remember that if we multiply or divide both sides of an inequality by a negative number, we must reverse the inequality sign.
To solve the inequality \(x - 4 > 0\), we would add 4 to both sides, yielding \( x > 4 \).
This means that for the function \ f(x) \ to be defined, x must be greater than 4.

Domain Constraints

The domain of a function is the set of all possible input values (x values) that allow the function to work.
For rational functions, domain constraints often arise from ensuring the denominator is not zero.
In this problem, the denominator is \( \sqrt{x-4} \), which must be greater than zero.
So, the expression inside the square root, \( x-4 \), must be positive, giving us the inequality \( x-4 > 0\).
Solving this inequality, we get \(x > 4 \).
Hence, to keep the function \( f(x) = \frac{x}{\sqrt{x-4}} \) defined, its domain must be all values of x that are greater than 4.
In interval notation, this domain is expressed as (4, ∞).
Ensuring these domain constraints is crucial to avoid mathematical errors and ensure the function behaves as expected.