Problem 76 - 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. $$ y=11 \sqrt{x} $$

Step-by-Step Solution

Answer

The domain of the function \( y = 11 \sqrt{x} \) is \( x \geq 0 \). By selecting values for x from this domain (0, 1, 4, 9, 16) and substituting them into the equation, the corresponding y values are found (0, 11, 22, 33, 44). The result is represented in form of a table.

Step 1: Identify Domain

The function is \( y = 11 \sqrt{x} \). For the square root, we needs to have \( x \geq 0 \). Therefore, the domain of the function is \( x \geq 0 \).

Step 2: Select Values

Now that we've identified the domain, we will select several values within this domain to create the table. A good starting point can be x = 0, then gradually incrementing the value of x. Here, let's select x = 0, x = 1, x = 4, x = 9 and x = 16.

Step 3: Compute y-values

Substitute each x value into the function to find the corresponding y value. So when x = 0, \( y = 11 \sqrt{0} = 0 \). When x = 1, \( y = 11 \sqrt{1} = 11 \). When x = 4, \( y = 11 \sqrt{4} = 22 \). When x = 9, \( y = 11 \sqrt{9} = 33 \). Finally, when x = 16, \( y = 11 \sqrt{16} = 44 \).

Step 4: Construct the Table

Now we'll construct a table that shows the x values in the domain and the corresponding y values. Our table will look like this:| x | y ||---|---|| 0 | 0 || 1 | 11|| 4 | 22|| 9 | 33|| 16| 44|

Key Concepts

Square root function

A square root function is a kind of mathematical function that involves the square root of the variable, most often denoted as \( \sqrt{x} \). In the context of our exercise, the function is written as \( y = 11 \sqrt{x} \), meaning the output, or \( y \), is determined by taking the square root of \( x \) and then multiplying it by 11.

Importantly, because you cannot take the square root of a negative number and have a real result, the domain of this function is restricted to values of \( x \) that are greater than or equal to zero. Hence, the domain is expressed as \( x \geq 0 \).

Understanding the constraints on the domain is crucial for exploring all the possible inputs that fit inside the function's rule and for creating accurate tables and graphs that represent our function.

Table of values

Creating a table of values is a helpful way to visualize how a function behaves. You choose specific values for \( x \) and then find the corresponding \( y \)-values using the function. This forms pairs of numbers that you can list as a table.

In this example, you start with \( x \) values within the domain identified in the previous step. We chose numbers like 0, 1, 4, 9, and 16, since they are easy to work with when taking square roots. By substituting these into our function \( y = 11 \sqrt{x} \), we find neat numbers for \( y \):

• \( x = 0 \rightarrow y = 0 \)
• \( x = 1 \rightarrow y = 11 \)
• \( x = 4 \rightarrow y = 22 \)
• \( x = 9 \rightarrow y = 33 \)
• \( x = 16 \rightarrow y = 44 \)

Recording these pairs provides a clear guide for understanding how changes in \( x \) affect \( y \), making predictions and more complex calculations more manageable.

Finding function values

Finding function values involves substituting specific numbers into your function in place of the variable and calculating the result. For the function \( y = 11 \sqrt{x} \), substituting each selected \( x \) value into the equation helps find the associated \( y \) value.

Let's explore a couple of these substitutions:

• For \( x = 0 \), substituting into the function gives \( y = 11 \sqrt{0} = 0 \).
• When \( x = 4 \), it results in \( y = 11 \sqrt{4} = 22 \).

Performing these substitutions for all chosen \( x \) values generates a complete picture of the function where you anticipate the results. These calculated values form the basis of the table of values and can help inform graphical representations or more in-depth analyses of the function's behavior. Learning how to perform these substitutions efficiently can increase your confidence when working with functions.