Problem 46 - Algebra and Trigonometry

Question

Find the domain of the function. $$ f(x)=\sqrt[4]{x+9} $$

Step-by-Step Solution

Answer

The domain of the function is \([-9, \infty)\).

Step 1: Understanding the Function

The function given is \( f(x) = \sqrt[4]{x+9} \). This is a fourth root function. We need to ensure that the expression under the root, \( x+9 \), is non-negative, because the fourth root is only defined for non-negative numbers in the set of real numbers.

Step 2: Setting up the Inequality

To find the domain, we set up an inequality for the expression under the root. We need \( x+9 \geq 0 \) to keep the fourth root defined as a real number. This means xx must be greater than or equal to -9.

Step 3: Solving the Inequality

Solve the inequality \( x+9 \geq 0 \) by subtracting 9 from both sides to isolate \( x \). This results in \( x \geq -9 \).

Step 4: Writing the Domain

The domain of the function is the interval where \( x \) satisfies the inequality. It can be written in interval notation as \([ -9, \, \infty )\), indicating that the function is defined for all \( x \) values from -9 to infinity.

Key Concepts

Fourth Root Function

When working with the fourth root function, it's important to know that it involves the expression \( f(x) = \sqrt[4]{x+9} \). A fourth root function is a type of radical function that evaluates the fourth root of a given expression. Because this is a real-valued function, we need to ensure that the expression inside the root, here \( x+9 \), is not negative. This is because you cannot take the fourth root of a negative number and get a real number as a result.

In general, for a function of the form \( \sqrt[n]{g(x)} \), where \( n \) is an even number like 4, the expression \( g(x) \) must be non-negative. In simpler terms:

• \( x+9 \) should be greater than or equal to zero to have a real result.
• This non-negativity condition ensures the domain covers values that don't lead to an undefined or imaginary output.

Understanding the nature of roots is crucial in determining domains. Fourth root functions, like square roots, are defined only for non-negative arguments.

Inequality Solving

Solving inequalities is a key skill needed to find the domain of functions involving roots and radicals. For the function \( f(x) = \sqrt[4]{x+9} \), we solve the inequality \( x+9 \geq 0 \) to determine where the function is valid.

The steps to solve this inequality are straightforward:

• First, recognize that the expression \( x+9 \) must be non-negative so the function stays defined as real.
• Subtract 9 from both sides of the inequality to isolate \( x \), resulting in \( x \geq -9 \).

This means that any value of \( x \) greater than or equal to \(-9\) will satisfy the condition, and the function can be evaluated using a real number.

Interval Notation

After solving the inequality, interval notation is used to express all the values that \( x \) can take in a concise format. For our function, after determining \( x \geq -9 \), we use interval notation \([ -9, \infty )\) to indicate the domain.

Interval notation is valuable for representing the set of all possible \( x \) values for which the function is defined:

• \([ -9, \infty )\) signifies that \( x \) includes \(-9\) and all numbers that are greater than \(-9\).
• The square bracket \([ \) next to \(-9\) denotes inclusion, meaning \(-9\) is part of the domain.
• The parenthesis \()\) next to infinity signals that infinity is not a specific number we can reach, but the values extend indefinitely towards it.

Interval notation thus serves as a simple yet powerful way to communicate the domain of a function.