Problem 45 - Algebra A Combined Function

Question

Without graphing, find the domain of each function. $$ g(x)=9-\sqrt{x+103} $$

Step-by-Step Solution

Answer

The domain of the function is \([-103, \infty)\).

Step 1: Understand the Function

The given function is \(g(x) = 9 - \sqrt{x + 103}\). This function involves a square root, which implies that the expression inside the square root must be non-negative in order for the function to be defined with real numbers.

Step 2: Set Up the Inequality

For the function to be defined, the expression inside the square root must satisfy \(x + 103 \geq 0\). This is necessary since the square root of a negative number is not a real number.

Step 3: Solve the Inequality

To find the values of \(x\) for which \(x + 103 \geq 0\), solve the inequality:\[x + 103 \geq 0\]Subtract 103 from both sides:\[x \geq -103\]

Step 4: Write the Domain

The domain of the function \(g(x)\) is all real numbers \(x\) that satisfy \(x \geq -103\). This can be written in interval notation as \([-103, \infty)\).

Key Concepts

Real Numbers

Real numbers are the most common numbers used in mathematics. They include all the numbers on the number line and are classified into several categories:

• Natural numbers like 1, 2, 3.
• Whole numbers which include 0 along with natural numbers.
• Integers which include both positive and negative whole numbers.
• Rational numbers which can be expressed as fractions.
• Irrational numbers which cannot be expressed as fractions, such as \(\pi\) and \(\sqrt{2}\).

Real numbers are critical as they are used to express measurements and continuous quantities. In the context of functions, especially those involving square roots, real numbers are necessary to define the domain. A square root function, for example, must produce a real result, which limits its domain to non-negative numbers for the expression under the square root sign. Understanding real numbers helps us determine where functions are defined and how they behave over the number line.

Square Root Functions

Square root functions feature an expression under a square root symbol, like \(\sqrt{x}\). This type of function can only have positive or zero values inside the square root to produce real numbers. If the expression were to be negative, the result would become a complex number, not real, which is often not desirable in basic function analysis.

Here's how the square root affects the domain:

• An expression like \(\sqrt{x}\) requires \(x \geq 0\).
• If the expression is \(\sqrt{x+a}\), then \(x+a\) must be at least 0, i.e., \(x \geq -a\).
• This principle extends to more complex expressions as long as the resulting value inside the square root is non-negative.

For our exercise function \(g(x) = 9 - \sqrt{x + 103}\), the domain constraint is \(x + 103 \geq 0\), implying \(x \geq -103\). This ensures we only consider input values that keep the function real.

Inequality

Inequalities are used to express a range of values that are permissible for a variable. They are critical when determining domains for functions, especially those involving square roots or other conditions.

To solve an inequality, it's essential to understand the symbols:

• \(>\) means greater than.
• \(<\) means less than.
• \(\geq\) means greater than or equal to.
• \(\leq\) means less than or equal to.

When working with inequalities, use operations such as addition or subtraction across the inequality to isolate the variable. However, remember that if you multiply or divide by a negative number, the inequality sign must be flipped.

In the exercise, solving the inequality \(x + 103 \geq 0\) involved subtracting 103 from both sides, resulting in \(x \geq -103\). This solution establishes that the function domain includes all x values starting from -103 and extending to infinity. Inequalities, therefore, help set the groundwork for understanding function constraints and acceptable input ranges.