Problem 4 - College Algebra and Calculus: An Applied Approach

Question

Determine if each value of \(x\) is in the domain of the expression.\(\sqrt{3 x-9} \quad\) (a) \(x=-3 \quad\) (b) \(x=3\)

Step-by-Step Solution

Answer

The value \(x = -3\) is not in the domain of the expression \(\sqrt{3x-9}\), while the value \(x = 3\) is in the domain of the expression.

Step 1: Define the Domain Condition

In any expression that includes a square root, the radicant must be greater than or equal to zero to yield a real number. Therefore, in the expression \(\sqrt{3x - 9}\), we have to ensure that \(3x - 9 \geq 0\) for the given \(x\) value to be in the domain of the expression.

Step 2: Check for Value \(x = -3\)

Substitute \(x = -3\) into the domain condition derived in Step 1: \(3(-3) - 9 = -18\). Since -18 is less than 0, \(x = -3\) is not in the domain of the expression.

Step 3: Check for Value \(x = 3\)

Substitute \(x = 3\) into the domain condition derived in Step 1: \(3(3) - 9 = 0\). Since the result is equal to 0, \(x = 3\) is in the domain of the expression.

Key Concepts

Square Root Function

Square roots are fascinating mathematical functions. They find the number which, when multiplied by itself, yields a specific value, referred to as the radicand. For instance, in the expression \( \sqrt{9} \), 9 is the radicand, and the square root is 3, since \( 3 \times 3 = 9 \).

In mathematics, square root functions often appear in expressions where we want to determine the values under the square root symbol that will result in real numbers. This is crucial because if we try to take the square root of a negative number in the real number system, we end up with imaginary numbers, not real ones. That's why we carefully define the conditions for the values within a square root expression, ensuring they result in meaningful, real outputs.

For example, in the expression \( \sqrt{3x - 9} \), it's crucial to identify all \( x \) values that make \( 3x - 9 \) non-negative. This helps us determine the domain, which we'll explore in the next section.

Domain Condition

Every function's domain is the set of input values (or \( x \) values) that will produce valid outputs. For square root functions, this means ensuring that the expression inside the square root, known as the radicand, is not negative. Ensuring this condition is met prevents outputs from becoming complex or imaginary numbers, which are beyond regular real value systems.

In our example, the expression is \( \sqrt{3x-9} \). Here, we apply the domain condition: \( 3x - 9 \geq 0 \). By solving this inequality, we determine when this condition is met:

• Add 9 to both sides: \( 3x \geq 9 \)
• Divide by 3: \( x \geq 3 \)

Thus, for any \( x \) greater than or equal to 3, the expression \( \sqrt{3x - 9} \) will be valid, remaining within the realm of real numbers. The domain here is \( x \geq 3 \), meaning all real numbers from 3 upwards.

Real Numbers

Real numbers are a fundamental part of mathematics. They include all numbers that can be found on the number line, encompassing whole numbers, positive and negative numbers, fractions, and irrational numbers like \( \sqrt{2} \).

For square root expressions, real numbers are crucial. If the radicand or the expression inside the square root becomes negative, the result is not a real number but an imaginary one. This is why ensuring the radicand remains non-negative establishes the boundaries of what's possible within real number constraints.

In the context of \( \sqrt{3x - 9} \), when the expression inside is zero or positive, it results in real numbers. However, if it turns negative, the results are no longer real, leading us into complex numbers. Thus, understanding and ensuring the domain helps maintain the expression's outcomes strictly in the real number space.