Problem 82 - Algebra and Trigonometry Real Mathematics, Real People

Question

Finding the Domain of a Function Find the domain of the function. $$g(x)=\sqrt{7-x}$$

Step-by-Step Solution

Answer

The domain of the function is (-∞,7].

Step 1: Setting up the inequality

We need to make sure that the value under the square root is non-negative, that is, we solve the inequality \(7 - x \geq 0\).

Step 2: Solving the inequality

To simplify the inequality, we can add x to both sides of the inequality to get \(7 \geq x\). This tells us that x can be any number that is less than or equal to 7.

Step 3: Express the domain

The domain of the function, therefore, includes all real numbers that are less than or equal to 7. This can be denoted in interval notation as (-∞,7], which means that the domain includes all numbers from negative infinity up to and including 7.

Key Concepts

Radical Function

A radical function is a type of function that includes a root symbol, which indicates root extraction, like a square root or a cube root. In the function \(g(x) = \sqrt{7-x}\), the square root is the radical. When working with radical functions, especially those with square roots, one crucial aspect to consider is ensuring the expression inside the square root is non-negative. This is because the square root of a negative number is not defined within the realm of real numbers. Thus, understanding the nature of radical functions lays the foundation for solving inequalities to find the domain of these functions. By identifying the conditions that make the function valid, we can determine the values of \(x\) that make the expression under the square root non-negative. This is the key to understanding radical functions.

Inequality Solving

Solving inequalities is a necessary skill when determining the domain of functions containing radicals. For the function \(g(x) = \sqrt{7-x}\), we start by setting up the inequality \(7-x \geq 0\). This ensures that the expression under the square root is non-negative. Solving the inequality stepwise is crucial for accuracy:

• Add \(x\) to both sides of the inequality: \(7 \geq x\).
• The result is \(x \leq 7\), meaning \(x\) can range from negative infinity up to, and including, 7.

Understanding this process helps clarify the restrictions on \(x\) that make the function valid and defined. This concept of translating mathematical conditions into inequalities and solving them is vital for gaining insight into where functions have real values. Inequality solving is a fundamental skill in algebra that allows us to find valid inputs for various types of functions.

Interval Notation

Interval notation is a convenient and concise way to describe a set of numbers, especially when dealing with inequalities. It allows us to represent the domain of a function without using lengthy sentences. For the function \(g(x) = \sqrt{7-x}\), after solving the inequality, we found that \(x\) can be any number that is less than or equal to 7.
This is expressed in interval notation as \((-\infty, 7]\), where:

• The parenthesis \((-\infty,\) indicates that the interval starts from negative infinity and does not include it, as infinity is a concept rather than a number.
• The bracket \(7]\) shows that the number 7 is included in the interval.

Using interval notation is not only a standard mathematical practice but also helps provide clarity and precision when communicating mathematical ideas. This notation is particularly useful when expressing the domain and range of functions clearly and concisely.