Question

Challenge Problem Find the domain of \(f(x)=\sqrt{\frac{x^{2}+1}{7-|3 x-1|}}\)

Short answer

The domain is \( (-2, \frac{8}{3}) \).

Step-by-step solution

Understand the function

The function given is \(f(x)=\sqrt{\frac{x^{2}+1}{7-|3x-1|}}\). The domain of this function includes all real numbers where the expression inside the square root is defined and non-negative.

Determine the square root constraint

For the square root \( \sqrt{A} \), the expression \( A \) inside the square root must be greater than or equal to 0. Here, \( \frac{x^{2}+1}{7-|3x-1|} \geq 0 \). Since \( x^2 + 1 \) is always positive for all \( x \), focus on \( 7 - |3x - 1| \) being positive to prevent division by zero.

Solve the inequality for the denominator

To ensure the function is defined, \( 7 - |3x - 1| > 0 \). This can be translated to \( 7 > |3x - 1| \).

Express the absolute value inequality

The inequality \( 7 > |3x - 1| \) means \( -7 < 3x - 1 < 7 \). To solve for \( x \), add 1 to all parts: \( -6 < 3x < 8 \).

Solve for x

Divide all parts of the inequality by 3: \( -2 < x < \frac{8}{3} \). Hence, the domain includes all \( x \) in the interval \( (-2, \frac{8}{3}) \).

Key concepts

square root function

A square root function involves taking the square root of an expression. For a function of the form \(f(x) = \sqrt{A}\), the value inside the square root, \(A\), must be non-negative because the square root of negative numbers isn't defined within the set of real numbers. Hence, when finding the domain of a square root function, you need to ensure the expression inside the square root is greater than or equal to zero.

The function provided, \(f(x) = \sqrt{\frac{x^{2} + 1}{7 - |3x - 1|}}\), requires that the entire fraction inside the square root remains non-negative at all times, and that the denominator is never zero, since division by zero is undefined.

absolute value inequality

An absolute value inequality is an inequality that contains the absolute value of a variable expression. The absolute value of an expression \(|A| \) is the distance of \A\ from zero on a number line, which is always non-negative and is defined as:

• \|A| = A \, if \A \geq 0
• \|A| = -A \, if \A < 0

In this problem, we have the inequality: \(7 - |3x - 1| > 0\). To solve it, we first isolate the absolute value expression: \(|3x - 1| < 7 \). This translates to: \-7 < 3x - 1 < 7 \. To solve for \x\, we first add 1 to all parts of the inequality: \-6 < 3x < 8 \, and then divide everything by 3 to isolate \x. This results in \-2 < x < \frac{8}{3}\.

Therefore, the solutions to the absolute value inequality are the values of \x\ that lie within the interval \(-2, \frac{8}{3}\).

real numbers

Real numbers include all the numbers that can be found on the number line. This covers both rational numbers (like 3, \frac{1}{2}\, and -4) and irrational numbers (like \sqrt{2}\ and \pi). When considering the domain of a function, it's important that the output values fall within the set of real numbers.

For the given function, \(f(x) = \sqrt{\frac{x^{2} + 1}{7 - |3x - 1|}}\), we note that \(x^{2} + 1 \) is always positive for all real values of \x\ because the square of any real number is non-negative, and adding one ensures that it is always positive.

Therefore, we don't have to worry about \x^{2} + 1\ falling outside the realm of real numbers. The real constraint comes from the absolute value inequality as explained earlier. Thus, the domain of our function, considering it needs to produce real numbers, consists of \(x \) values in the interval \(-2, \frac{8}{3}\).