Problem 16 - Algebra 1

Question

Make an input-output table for the function. Use 0, 1, 2, and 3 as the domain. $$ y=6 x+1 $$

Step-by-Step Solution

Answer

The input-output table for the function \(y=6x+1\) is as follows; for \(x=0\), \(y=1\); for \(x=1\), \(y=7\); for \(x=2\), \(y=13\); and for \(x=3\), \(y=19\).

Step 1: Identify function and domain

The function in this case is \(y = 6x + 1\). The domain of the function, which are the specific set of x-values we are asked to use, are \(x = 0, 1, 2, 3\).

Step 2: Compute output for each input

Use the given function to compute the corresponding y-value for each x-value in the domain. 1. For \(x = 0\), \(y = 6 \times 0 + 1 = 1\). 2. For \(x = 1\), \(y = 6 \times 1 + 1 = 7\). 3. For \(x = 2\), \(y = 6 \times 2 + 1 = 13\). 4. For \(x = 3\), \(y = 6 \times 3 + 1 = 19\).

Step 3: Fill in the function table

Based on the outputs from Step 2, we can fill in the table. | x | y | |---|---| | 0 | 1 | | 1 | 7 | | 2 | 13| | 3 | 19|

Key Concepts

Functions

Functions are one of the fundamental concepts in mathematics. They relate an input to an output according to a specific rule or formula. In our exercise, the function is expressed as a linear equation: \[ y = 6x + 1 \]. This means for every value of \( x \) that you substitute into the equation, you get a unique \( y \) value back. The equation specifies exactly how \( x \) and \( y \) are related.

When using functions, the process generally involves the following steps:

• Identify the function you are working with, which tells you the rule for calculating outputs.
• Determine the domain, which is the set of input values you will use.
• Calculate the corresponding output for each input using the function's rule.

Understanding and working with functions help solve various real-life problems, such as predicting future trends, modeling physical phenomena, and more.

Domain and Range

The domain and range are crucial concepts in understanding functions. The domain represents all the possible input values, or \( x \) values, that can be fed into a function. In our problem, the domain is specified as \( x = 0, 1, 2, 3 \). These are the inputs for which we will find corresponding outputs.

The range, on the other hand, is all the possible output values, or \( y \) values, that you get after applying the function to the domain. For the function \( y = 6x + 1 \), we calculated:

• When \( x = 0\),\( y = 1\)
• When \( x = 1\),\( y = 7\)
• When \( x = 2\),\( y = 13\)
• When \( x = 3\),\( y = 19\)

So, the range of the function is \( y = 1, 7, 13, 19 \).

Understanding domain and range is key to knowing the scope and limits of any function. It helps you see the operation and constraints within which the function operates.

Linear Equations

Linear equations are a type of function characterized by a straight line when graphed. They have a constant rate of change, making them predictable and easy to work with. The general form of a linear equation is \( y = mx + b \), where:

• \( m \) is the slope, representing the change in \( y \) for a change in \( x \).
• \( b \) is the y-intercept, where the line crosses the y-axis.

In the exercise problem, the linear equation is \( y = 6x + 1 \), meaning:

• The slope \( m = 6 \), indicates that for every unit increase in \( x \), \( y \) increases by 6 units.
• The y-intercept \( b = 1 \), is where the line crosses the y-axis, at y = 1 when \( x = 0 \).

Linear equations are straightforward since their graph is always a straight line. They are foundational in modeling relationships where there is a constant rate of change, such as speed, cost, or other linear phenomena.