Problem 61 - Algebra 1

Question

Make an input-output table for the function \(A=8+2.5 t\) when \(t=2,3,4,5,\) and \(6 .\) Describe the domain and the range of the function whose values are shown in the table.

Step-by-Step Solution

Answer

The input-output table of the function will be: for \(t=2\), \(A=13\); \(t=3\), \(A=15.5\); for \(t=4\), \(A=18\); for \(t=5\), \(A=20.5\); for \(t=6\), \(A=23\). The domain of the function is \(t = 2, 3, 4, 5, 6\) and the range is \(A = 13, 15.5, 18, 20.5, 23\).

Step 1: Create an Input-Output Table

Substitute the given values for \(t\) into the function \(A=8+2.5 t\) to find the corresponding value of \(A\).For \(t=2\), compute \(A\) as: \(A=8+2.5*2=13\)For \(t=3\), compute \(A\) as: \(A=8+2.5*3=15.5\)For \(t=4\), compute \(A\) as: \(A=8+2.5*4=18\)For \(t=5\), compute \(A\) as: \(A=8+2.5*5=20.5\)For \(t=6\), compute \(A\) as: \(A=8+2.5*6=23\)

Step 2: Determine the Domain

The domain of a function is the set of all possible input values (here, \(t\) values). In this case, the domain is \(t = 2, 3, 4, 5, 6\)

Step 3: Identify the Range of the Function

The range is the set of all possible outputs of the function, in this case, corresponding \(A\) values. For the given values of \(t = 2, 3, 4, 5, 6\), the calculated range is \(A = 13, 15.5, 18, 20.5, 23\)

Key Concepts

Input-Output Tables

An input-output table is a simple yet powerful tool used in algebra to show how a function links input values with their corresponding output values. It's like a translator between numbers, helping you visualize the relationship between variables.When you have a function such as \(A = 8 + 2.5t\), the input-output table can be created by substituting specific values into the formula. Imagine you've chosen the values \(t = 2, 3, 4, 5,\) and \(6\). Plug these inputs into the function to find the outputs.

• For \(t = 2\), substitute into the function: \(A = 8 + 2.5 \times 2 = 13\).
• For \(t = 3\), substitute into the function: \(A = 8 + 2.5 \times 3 = 15.5\).
• This continues for each of your selected inputs.

This step-by-step substitution shows the output values associated with each input. Using the input-output table allows you to observe how changes in input \(t\) affect the output \(A\). It simplifies understanding changes and patterns in functions.

Domain and Range

When discussing functions, the terms "domain" and "range" quickly come into play. They describe the limits of the function in terms of inputs and outputs.The domain is all the possible values you can choose for the input. In our earlier example, the domain consists of the values where \[ t = \{2, 3, 4, 5, 6\}\]. These are the chosen numbers allowed within the function for \(t\).On the flip side, the range refers to all the possible outputs that the function can produce. In this scenario, when you plug the domain values into the function \(A = 8 + 2.5t\), you get the range: \[ A = \{13, 15.5, 18, 20.5, 23\}\].By clearly defining the domain and range, you not only determine the scope of your function but also allow for a deeper understanding of its behaviour. Knowing these terms helps in visualizing the function on a graph and can simplify solving more complex algebra problems.

Linear Functions

Linear functions, a fundamental concept in algebra, are those whose graph forms a straight line. They have the basic form \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept.In our case, the function \(A = 8 + 2.5t\) fits this definition with \(A\) as the output, \(t\) as the input variable, \(2.5\) as the slope, and \(8\) as the y-intercept. Here, you can visualize it as moving up the y-axis by \(2.5\) units for each 1 unit increase along the x-axis (or in this case, the \(t\)-axis).A linear function translates neatly into a clear pattern:

• Slope of the line \(= 2.5\)
• Y-intercept \(= 8\)
• Pattern of change is constant

Understanding the form and components of a linear function aids in predicting the future behavior of the data. This predictability is why linear functions are so widely used in various applications, from business to science, providing a foundation for more advanced topics in mathematics.