UpStudy Free Solution:

To find the range of the function \(y = - 3x + 1\) for the given domain \(\{ 2, 8\} \), we need to evaluate the function at each value in the domain.

1. For \(x = 2\):

\[y = - 3( 2) + 1 = - 6 + 1 = - 5\]

2. For \(x = 8\):

\[y = - 3( 8) + 1 = - 24 + 1 = - 23\]

So, the range of the function \(y = - 3x + 1\) for the domain \(\{ 2, 8\} \) is \(\{ - 5, - 23\} \).

Supplemental Knowledge

Understanding how to evaluate functions and determine their range for a given domain is a fundamental skill in algebra. This process involves substituting each value from the domain into the function and calculating the corresponding output values.

Evaluating Linear Functions:

A linear function is typically written in the form \(y = mx + b\), where:

- \(m\) is the slope of the line.

- \(b\) is the y-intercept.

To evaluate a linear function for specific values:

1. Substitute Each Domain Value into the Function:

- Replace \(x\) with each value from the domain.

2. Calculate the Corresponding Output:

- Perform arithmetic operations to find \(y\).

For example, consider evaluating \(y = 2x - 3\) for \(x = 1, 4, 7\):

- For \(x = 1\):

\[y = 2( 1) - 3 = 2 - 3 = - 1\]

- For \(x = 4\):

\[y = 2( 4) - 3 = 8 - 3 = 5\]

- For \(x = 7\):

\[y = 2( 7) - 3 = 14- 3= 11\]

The range of this function for the given domain is \(\{ - 1,5,11\} \).

Finding Range:

The range of a function consists of all possible output values (y-values). When given a specific domain (set of x-values), you can determine the range by evaluating each element in that domain.

Receptively understanding how to evaluate functions and determine their ranges can significantly hone your mathematical proficiency! For further assistance with these concepts or any other complex topics in math or science, UpStudy is always ready and waiting! Don't let difficult homework slow your learning experience - join UpStudy today and maximize it!