Problem 75 - A Graphical Approach to College Algebra

Question

Determine the domain of each function. Do not use a calculator. $$f(x)=-\sqrt[4]{6-x}$$

Step-by-Step Solution

Answer

The domain is \\( (-\infty, 6] \\\).

Step 1: Understanding the Function

The function given is the fourth root of an expression, specifically \( f(x) = -\sqrt[4]{6-x} \). Fourth roots are defined for non-negative numbers, so the expression inside the root, \( 6-x \), must be non-negative.

Step 2: Set Non-Negative Condition

To find when \( 6-x \) is non-negative, we set up the inequality \( 6-x \geq 0 \). This will give us the allowable values of \( x \) that make the expression inside the root defined.

Step 3: Solve the Inequality

Solving \( 6-x \geq 0 \) involves adding \( x \) to both sides to get \( 6 \geq x \) or equivalently, \( x \leq 6 \). This indicates that \( x \) must be less than or equal to 6.

Step 4: Determine the Domain

Since \( x \leq 6 \), the domain of the function is all real numbers less than or equal to 6. In interval notation, this is expressed as \( (-\infty, 6] \).

Key Concepts

Inequalities

When dealing with functions, especially those involving roots or other restrictions, inequalities are a vital tool. They help us figure out which values are permissible for a variable. An inequality like \( 6-x \geq 0 \) indicates that the expression \( 6-x \) should not be negative if the fourth root is to exist.
To solve this, you need to isolate \( x \) on one side. Add \( x \) to both sides and you're left with \( 6 \geq x \), or just \( x \leq 6 \). This inequality gets us one step closer to describing the domain of a function by highlighting what \( x \) values keep the function defined.
Inequalities often come up in mathematics when you need to ensure particular conditions are true. These could be conditions of positivity, non-negativity, or other relational requirements, such as those arising from square roots, logarithms, or division where denominators can't be zero.

Fourth Root

The fourth root of a number is a value that, when multiplied by itself four times, gives back the original number. In mathematical terms, for a number \( y \), \( \sqrt[4]{y} \) represents this root. It's similar to the square root, but requires multiplying by itself an additional two times.
For our specific function, \( f(x) = -\sqrt[4]{6-x} \), the expression \( 6-x \) must be non-negative because any root, including the fourth, is only defined for non-negative numbers under real numbers. This is why solving the inequality \( 6-x \geq 0 \) is crucial.
The minus sign in front of the root doesn't affect the need for \( 6-x \) to be non-negative. It merely flips the sign of the output once the root is calculated. Understanding this helps establish the function's domain and ensures the square root part is well-defined and solvable under real numbers.

Interval Notation

Interval notation is a shorthand used in mathematics to describe sets of numbers, particularly ranges of possible values for variables. When we've determined that \( x \leq 6 \), we can use interval notation to express this range more succinctly.
The domain \( x \leq 6 \) translates into interval notation as \( (-\infty, 6] \). Here, \(-\infty\) means there's no lower limit on the values \( x \) can take – it can be any negative number. The square bracket around 6, \([6]\), signifies inclusion of 6 in the domain.
Interval notation is not only about brevity. It visually helps indicate whether endpoints of the interval are included or not. Round parentheses \(()\) indicate exclusion (like \(-\infty\)), while square brackets \([]\) indicate inclusion (like \(6\)). Understanding this notation is vital in calculus, analysis, and various areas of higher mathematics.