Problem 78 - Algebra 1: Concepts and Skills

Question

Find the domain of the function. Then use several values in the domain to make a table of values for the function. $$ y=\sqrt{x-8} $$

Step-by-Step Solution

Answer

The domain of the function \(y = \sqrt{x - 8}\) is \(x \geq 8\). A table of values includes (8, 0), (9, 1), (10, \(\sqrt{2}\)), (16, \(\sqrt{8}\)), and (25, \(\sqrt{17}\)).

Step 1: Finding the Domain

The domain of a function encompasses all possible x values. Considering the function has a square root, the value inside the square root must be greater than or equal to zero (as we can't take the square root of a negative number in the real number system). This means that \(x-8 \geq 0\) . Solving this inequality will give the domain of the function.

Step 2: Solving the Inequality

To solve this inequality \(x - 8 \geq 0\), add 8 to both sides, which gives the inequality \(x \geq 8\). So, this is the domain of the function.

Step 3: Creating a Table of Values

Choosing several x values greater than or equal to 8 (let's choose 8, 9, 10, 16, and 25), the corresponding y values can be calculated using the given function \(y = \sqrt{x-8}\). To find the y values, substitute the chosen x values into the function and calculate the square root. For \(x = 8\), \(y = \sqrt{8 - 8} = 0\). For \(x = 9\), \(y = \sqrt{9 - 8} = 1\). For \(x = 10\), \(y = \sqrt{10 - 8} = \sqrt{2}\). For \(x = 16\), \(y = \sqrt{16 - 8} = \sqrt{8}\). For \(x = 25\), \(y = \sqrt{25 - 8} = \sqrt{17}\). Thus, a table of values has been calculated.

Key Concepts

Understanding Inequalities

When solving for the domain of functions involving square roots, inequalities often come into play. An inequality is essentially a statement about the relative size or order of two values. In math, you'll frequently see symbols like \(<, >, \leq,\) and \(\geq \). These symbols tell us how two expressions compare without stating that they are equal. For the function \(y=\sqrt{x-8}\), you need to ensure that the expression under the square root is never negative. That's why we use the inequality \(x-8 \geq 0\).

• The symbol \(\geq\) denotes 'greater than or equal to'. This tells us that \(x-8\) must be zero or any value above zero.
• Solving the inequality means finding all the x-values that make the inequality true. Here, adding 8 to both sides of \(x-8 \geq 0\) gives us \(x \geq 8\).
• Solving inequalities is a step-by-step process, where you often add, subtract, multiply, or divide both sides by the same number to isolate the desired variable (in this case \(x\)).

Adhering to these rules ensures that the function remains defined for all chosen x-values.

Exploring Square Root Functions

Square root functions, such as \(y=\sqrt{x-8}\), have specific characteristics that are essential to understand, especially when determining a function's domain. Generally, a square root function involves finding a number that, when multiplied by itself, gives the original number under the square root.

• The square root function typically results in real numbers only for non-negative values of the expression under the root.
• This is why inequalities play a crucial role in determining the valid x-values (domain) for which the function is defined.
• For our function, the expression \(x-8\) inside the square root must be zero or positive for \(y\) to be a real number.

Remember, square root functions usually start at the point, where their expression becomes zero, and onwards, reflecting their gradual increase as the x-value grows. This fundamental understanding helps predict the graph's shape and behavior of such functions.

Introduction to Real Numbers

In mathematics, real numbers encompass a vast and significant set of numbers we often use in everyday life. These include rational numbers like integers, fractions, and even irrational numbers like \(\sqrt{2}\) and \(\pi\). Real numbers are intrinsic to understanding domains and ranges in functions.

• When dealing with functions like \(y=\sqrt{x-8}\), we're concerned with ensuring all results fall within the real number category.
• The use of inequalities ensures that the results remain real by guaranteeing that the square root's argument is non-negative.
• Negative arguments inside a square root would lead us into complex numbers, which are beyond the scope of real numbers.

This restriction to non-negative values for square roots is what keeps functions like ours within the real, manageable realm. Sticking to real numbers simplifies calculations, making functions easier to understand and graph.