Problem 88 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the domain of \(x\) in the expression. $$\sqrt{6 x+15}$$

Step-by-Step Solution

Answer

The domain of the function \(f(x) = \sqrt{6x + 15}\) is \(x \geq -2.5\)

Step 1: Determine the condition for the radicand

Make the expression inside the square root, \(6x + 15\), greater or equal to zero. This gives you the inequality \(6x + 15 \geq 0\).

Step 2: Solve the inequality

Subtract 15 from both sides of the inequality to isolate \(6x\). This yields \(6x \geq -15\). As a next step, divide both sides by 6 to solve for \(x\). This gives us the inequality \(x \geq -15/6 \).

Step 3: Simplify the solution

After simplifying the fraction we obtain \(x \geq -2.5\).

Key Concepts

Inequalities

Understanding inequalities is fundamental when finding the domain of a function. An inequality is a mathematical statement that one value is less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥) another. In the context of finding domains, we use inequalities to express the range of values for which the function is defined. When we are presented with a function that includes a square root, we set the radicand (the expression under the square root) to be greater than or equal to zero and solve for the variable to find the domain.

For the problem at hand, the inequality in Step 1, \(6x + 15 \geq 0\), is solved by isolating the variable 'x'. This is done by performing operations on the entire equation, such as subtracting 15 and then dividing by 6, while maintaining the inequality relationship. Once the variable is isolated, we get the domain of 'x' where the original equation is valid, meaning \(x \geq -2.5\) as found in Step 3.

Radicals

Radicals, or roots, are another key concept in algebra, particularly when determining the domain of functions that include them. A radical function includes a variable under a root sign. The most common type is a square root, but cubic roots, fourth roots, and higher are also possible. The principal thing to remember with radicals is that the radicand (the number or expression inside the radical) must be non-negative when dealing with square roots, since negative numbers do not have real-number square roots.

In our exercise, we handle a square root which imposes a restriction on our domain. The square root in the expression \(\sqrt{6x+15}\) specifies that \(6x+15\) must be greater than or equal to zero. This restriction ensures that the values plugged into 'x' will not result in taking the square root of a negative number, which is undefined in the set of real numbers.

Domain in Algebra

The domain of a function is the set of all possible input values (typically represented by 'x') that the function can accept without resulting in any undefined or non-real numbers. In algebra, finding the domain often involves identifying restrictions on the variables that could lead to undefined expressions, such as division by zero or square roots of negative numbers. The steps we follow to find the domain can include setting up and solving inequalities, as mentioned above.

When we plug \(x \geq -2.5\) into the original function, it will yield only real numbers, ensuring that the domain is correctly identified. In real-life contexts, domains can reflect practical limitations, such as physical dimensions that cannot be negative, or quantities that must be within a certain range.

Square Roots

The square root function, represented as \(\sqrt{x}\), is a special type of radical function where the index (the small number above and to the left of the root symbol) is 2. This function yields a positive number which, when multiplied by itself, gives the original number under the square root. A fundamental aspect of square roots is that they are not defined for negative numbers in the realm of real numbers, which is why we require the radicand to be non-negative when finding a function's domain.

For the given exercise, the square root of \(6x+15\) necessitates that \(6x+15\) must be at least zero, which is reflected in the inequality that we solve to find the domain. This step ensures that the outputs of the square root are real numbers, and hence the function is well-defined across its domain.