Problem 20 - Algebra 1

Question

Make an input-output table for the function. Use 1, 1.5, 3, 4.5, and 6 as the domain. $$ y=32-3 x $$

Step-by-Step Solution

Answer

Input-output table: {(1, 29), (1.5, 27.5), (3, 23), (4.5, 18.5), (6, 14)}

Step 1: Substitution for x = 1

We substitute x = 1 into the function to find the corresponding y value: \(y = 32 - 3 * 1\)

Step 2: Substitution for x = 1.5

Similar to the previous step, now substitute x = 1.5 into the function: \(y = 32 - 3 * 1.5\)

Step 3: Substitution for x = 3

Substitute x = 3 into the function: \(y = 32 - 3 * 3\)

Step 4: Substitution for x = 4.5

Substitute x = 4.5 into the function: \(y = 32 - 3 * 4.5\)

Step 5: Substitution for x = 6

Finally, substitute x = 6 into the function: \(y = 32 - 3 * 6\)

Step 6: Compile the Results

To finish, compile the pairs of x and y values that you computed in the previous steps. These will form your input-output table.

Key Concepts

Input-Output Tables

Input-output tables are useful for visualizing how changes in input affect the output based on a specific function. In this example, the function given is a linear one: \(y = 32 - 3x\). An input-output table helps organize data by listing inputs (also known as the domain) alongside their corresponding outputs (the range).
Creating the table involves the following steps:

• Choose a set of x-values, which will be the domain of the function. Here, the domain is {1, 1.5, 3, 4.5, 6}.
• Substitute each x-value into the function to compute the corresponding y-value.
• Fill in the table with x-values in one column and their computed y-values in another column.

This method offers a straightforward way to see the relationship between different values of x and their corresponding y, helping students grasp the concept of functions and how they work.

Domain and Range

Understanding the domain and range of a function is crucial when dealing with mathematical functions. For any function, the domain refers to the set of all possible input values, in this case, the x-values. The range is the set of all possible output values, or y-values, that result from substituting the domain values into the function.
For the given linear function \(y = 32 - 3x\), you can define:

• The domain as the specific values provided: {1, 1.5, 3, 4.5, 6}.
• The range consists of the y-values obtained by substituting these x-values into the function.

Identifying domain and range is essential because it frames the limits within which a function operates. This understanding helps in predicting and describing function behavior.

Substitution Method

The substitution method is a technique used to solve functions by replacing variables with known values to find unknowns. In this context, you substitute each x-value from the domain into the function to calculate the corresponding y-value.
Follow these steps to apply this method:

• Take a value from the domain (e.g., x = 1, 1.5, etc.).
• Insert the value into the given function (e.g., substituting x = 1 gives \(y = 32 - 3(1)\)).
• Compute the result to find y.
• Repeat for all domain values to create a complete input-output table.

This method is particularly helpful for linear functions because it systematically reveals how each input affects the output. By practicing substitution, students solidify their understanding of how linear equations operate in practical scenarios.