Problem 13 - Algebra 1

Question

Make an input-output table for the function. Use 0, 1, 2, and 3 as the domain. $$ y=3 x+2 $$

Step-by-Step Solution

Answer

The input-output table for the function \(y=3x+2\) with domain values 0, 1, 2, 3 is: \n | x | y |\n|---|---|\n| 0 | 2 |\n| 1 | 5 |\n| 2 | 8 |\n| 3 | 11 |

Step 1: Understanding the Function

We have a linear function here, \(y=3x+2\), where 'x' is the input and 'y' is the output. For each input value of 'x', there will be corresponding output value for 'y' which can be calculated by substituting 'x' in the function.

Step 2: Calculating Output for First Input Value

The first input value is 0. Substitute this value in the function. So, \(y=3(0)+2=2\). Thus, when \(x=0\), \(y=2\). This gives us our first pair in the input-output table.

Step 3: Calculating Output for Second Input Value

Now, let's use the second input value which is 1. Substituting in the function gives: \(y=3(1)+2=5\). Thus when \(x=1\), \(y=5\). Now we have our second pair.

Step 4: Repeating the Process for Other Input Values

Repeat the process for the remaining input values of 2 and 3. Substitution gives: For \(x=2\), \(y=3(2)+2=8\) and for \(x=3\), \(y=3(3)+2=11\).

Step 5: Summarizing the Data in a Table

Finally, we can construct our input-output table using the computed values: \n| x | y |\n|---|---|\n| 0 | 2 |\n| 1 | 5 |\n| 2 | 8 |\n| 3 | 11 |

Key Concepts

Input-Output Table

An input-output table is a convenient way to organize information about a function’s inputs and their corresponding outputs. Think of it like a table that helps us map out values. The input or 'x' will be listed in one column, and the corresponding output or 'y' will be in the next. This table is incredibly helpful when dealing with linear functions like the one in our exercise.
For the function given, which is defined by the equation \( y = 3x + 2 \), we plot the inputs—0, 1, 2, and 3—into the function to find the respective outputs. By following a step-by-step calculation:

• For \( x = 0 \), \( y = 3(0) + 2 = 2 \)
• For \( x = 1 \), \( y = 3(1) + 2 = 5 \)
• For \( x = 2 \), \( y = 3(2) + 2 = 8 \)
• For \( x = 3 \), \( y = 3(3) + 2 = 11 \)

By documenting these pairs into the table, we establish a clear relation between each input and its resulting output. This provides a visual aid that goes hand-in-hand with understanding the behavior of the function.

Domain and Range

In the context of functions, the terms "domain" and "range" define the boundaries of the function. The domain consists of all the possible input values (x-values) that the function can accept. In our exercise, the specified domain is 0, 1, 2, and 3. These values are what we'll substitute into the function equation.
Once we know the domain, we can determine the range. The range is the set of all possible outputs (y-values) that correspond to the domain inputs through the function relationship. For the equation \( y = 3x + 2 \), after substituting the domain values:

• Input \( x = 0 \) results in output \( y = 2 \)
• Input \( x = 1 \) results in output \( y = 5 \)
• Input \( x = 2 \) results in output \( y = 8 \)
• Input \( x = 3 \) results in output \( y = 11 \)

Thus, the range of our function is \( \{2, 5, 8, 11\} \). Understanding the domain and range conceptually helps in grasping the extent of the function's capability and visualizing the scope of outputs.

Function Evaluation

Function evaluation involves computing the output (y-value) of a function for a given input (x-value). It requires substituting the input into the equation to determine the corresponding output.
To evaluate the function \( y = 3x + 2 \) for a specific input, we replace 'x' in the equation with the chosen input value:

• For example, with input \( x = 0 \), substitute to get \( y = 3(0) + 2 = 2 \)
• Similarly, input \( x = 1 \) yields \( y = 3(1) + 2 = 5 \)
• Continuing with \( x = 2 \), compute \( y = 3(2) + 2 = 8 \)
• Finally, for \( x = 3 \), find \( y = 3(3) + 2 = 11 \)

This process is straightforward—plug in the input value for 'x' and solve the expression to find 'y'. Function evaluation is essential for understanding how a function operates and for making calculations regarding any given input within the defined domain.