Problem 9 - College Algebra

Question

For the following exercises, find the domain of each function using interval notation. $$ f(x)=3-\sqrt{6-2 x} $$

Step-by-Step Solution

Answer

The domain of the function is \((-\infty, 3]\).

Step 1: Identify the Expression Inside the Square Root

The domain of the function will be defined based on the values that make the expression inside the square root non-negative. The expression inside the square root is \(6 - 2x\). The square root function is defined only for non-negative numbers, so we need \(6 - 2x \geq 0\).

Step 2: Solve the Inequality for \(x\)

To find the values of \(x\) that satisfy \(6 - 2x \geq 0\), solve the inequality by isolating \(x\): \[6 - 2x \geq 0\] Subtract 6 from both sides: \[-2x \geq -6\] Divide both sides by \(-2\), noting that you must reverse the inequality sign: \[x \leq 3\]

Step 3: Express the Domain in Interval Notation

The solution to the inequality \(x \leq 3\) shows that \(x\) can take any value less than or equal to 3. Therefore, the domain of \(f(x)\) in interval notation is \((-\infty, 3]\).

Step 4: State the Domain

The domain of the function is all real numbers \(x\) such that \(x\) is less than or equal to 3. This ensures that the square root is always defined. Thus, the domain is \((-\infty, 3]\).

Key Concepts

Interval Notation

In mathematics, interval notation offers a concise way to represent a set of values that belong to a specific segment of the real number line. This can include all numbers between two endpoints, with specific rules about whether these endpoints are included. For example, if an endpoint is included, we use a square bracket \([\), whereas a round bracket \((\) indicates it's not included. Here’s a quick breakdown:

• \([a, b]\): Includes all numbers from \(a\) to \(b\), and both points are included.
• \((a, b)\): Includes numbers between \(a\) and \(b\), but not the endpoints themselves.
• \((a, b]\) or \([a, b)\): One endpoint is included, the other is not.

The domain of a function is ideally expressed in interval notation. When considering domains, it’s crucial to establish which intervals make a function valid or invalid. Anything outside this region may not fit within the domain under normal mathematical assumptions.

Square Root Function

The square root function is an essential concept in mathematics. It is the inverse operation of squaring a number. The defining feature of a square root function is that it only produces non-negative outputs. This characteristic influences the function's domain significantly. Let's dive into the basic properties:

• The square root function, denoted by \(\sqrt{a}\), returns the non-negative root of \(a\).
• For a real-valued output, the input (under the square root) must be non-negative, meaning \(a \geq 0\).

This translates directly to domain constraints since any negative number under a square root in real numbers could result in an undefined outcome. Therefore, when solving a problem involving square roots, make sure the expression is set to be greater than or equal to zero, confirming the values of input produce real outputs.

Inequalities

Inequalities are mathematical expressions that establish the relative size or order of two values. They can look simple at a glance but are a powerful tool in determining a function’s domain. Here are the basic inequality symbols:

• \(>\) and \(<\): Greater than, less than respectively.
• \(\geq\) and \(\leq\): Greater than or equal to, less than or equal to.

When working with inequalities in a domain context, you often need to solve them to see which values are permissible for function usage. For instance, if an inequality shows \(6 - 2x \geq 0\), you need to transform it to find where it holds true.
Here's how you achieve that:

• Isolate the variable, as done with equations.
• If you multiply or divide by a negative number, switch the inequality sign.
• Once solved, use this result to express the domain accurately in interval notation.

Inequalities thus set precise constraints and help us find valid inputs that help sustain function integrity.