Problem 31 - Algebra 1: Concepts and Skills

Question

Find the domain of the function. Then use severalvalues in the domain to make a table of values for the function. $$y=\sqrt{3 x-10}$$

Step-by-Step Solution

Answer

The domain of the function \(y = \sqrt{3x-10}\) is \([3.33, ∞)\). A table of values for the function includes (3.33, 0), (4, 1.15), and (5, 1.82).

Step 1: Find the Domain

The crucial step is to determine the domain of the function. Since we have a square root, and we can't take the square root of a negative number, the domain of the function includes all x-values for which \(3x - 10 \geq 0\). So, solve for x in the inequality \(3x - 10 \geq 0\). Adding 10 to both sides of the inequality gets \(3x \geq 10\). Therefore, \(x \geq \frac{10}{3}\), which is approximately 3.33. So, the domain is \([3.33, ∞)\)

Step 2: Choose Values and Calculate y

Now we select some x-values from the domain we just found to create a table of values. Let's choose 3.33 (the lower bound of our domain), 4, and 5. Plug these values individually into the function \(y = \sqrt{3x - 10}\) to calculate the corresponding y-values.

Step 3: Tabulate the Values

Substitute the selected x values into the function to find the corresponding y-values. When \(x=3.33\), \(y= \sqrt{3(3.33) - 10} = 0\). When \(x=4\), \(y= \sqrt{3(4) - 10} \approx 1.15\). And, when \(x=5\), \(y= \sqrt{3(5) - 10} \approx 1.82\). So the table of values is: \\[\begin{align*}x & y \3.33 & 0 \4 & 1.15 \5 & 1.82 \\end{align*}\]

Key Concepts

Function Tables

Function tables are a great way to visually represent the input-output relationships of a function. In essence, they allow us to see how different inputs (x-values) affect the corresponding outputs (y-values). This is especially helpful in understanding the behavior of a function across its domain. Let's break it down:

• First, identify the domain of the function. This represents all the valid x-values you can plug into the function.
• Next, select some actual x-values from this set. It is usually helpful to choose a few values, including boundary points and easily computable points.
• Finally, calculate the corresponding y-values by substituting these x-values into the function. This gives us a table showing our inputs and their resulting outputs.

By using function tables, we gain insight into the function's behavior, helping us understand concepts like continuity, increasing/decreasing intervals, and more.

Square Root Function

The square root function is a specific type of mathematical function that involves finding the non-negative square root of a number. It is typically written as \( y = \sqrt{x} \), but can include other expressions inside the square root, like \( y = \sqrt{3x - 10} \) from the original exercise.

• A key characteristic of the square root function is that it is only defined for numbers inside the square root that are greater than or equal to zero. This means you have to solve an inequality to determine its domain.
• The graph of a basic square root function \( y = \sqrt{x} \) is a curve that starts at the origin (0,0) and rises slowly to the right, only staying in the first quadrant.
• If there is a more complex expression inside the square root, like \( 3x - 10 \), you solve \( 3x - 10 \geq 0 \) to find the domain.

Understanding the behavior of the square root function helps us accurately determine valid x-values and predict the shape of its graph.

Inequalities in Algebra

In algebra, inequalities are expressions involving the symbols \( > \), \( < \), \( \geq \), or \( \leq \). They are used when we want to express a range of possible values rather than a single solution.

• To solve an inequality, treat it much like an equation. However, remember that if you multiply or divide by a negative number, the inequality sign flips.
• The solution set to an inequality represents a range of values. For example, solving \( 3x - 10 \geq 0 \) gives us \( x \geq \frac{10}{3} \).
• Often, inequalities help us determine the domain of functions like the square root function, indicating which x-values lead to a real number output.

Mastering inequalities in algebra is critical, as they are fundamental in expressing constraints and conditions in various mathematical contexts.