- We identify the domain as \(x \in \{0, 2, 4, 6\}\).
- We substitute each value of \(x\) into the function \(y = \frac{1}{2}x + 1\).
- We compute the corresponding \(y\)-values for each \(x\).
- The resulting ordered pairs are \(\boxed{\{(0, 1), (2, 2), (4, 3), (6, 4)\}}\).
Explanation
- Understanding the Problem and Data Analysis
The problem asks us to evaluate the linear function \(y = \frac{1}{2}x + 1\) for a specific set of input values, or the domain, which is \(\{0, 2, 4, 6\}\). A linear function is a mathematical relationship where the output \(y\) changes at a constant rate with respect to the input \(x\). To find the corresponding output values (or range), we substitute each element of the domain into the function expression. We will calculate the y-coordinate for each given x-coordinate to form ordered pairs \((x, y)\).
- Evaluating the Function for the Given Domain
We evaluate the function \(y = \frac{1}{2}x + 1\) for each x-value in the domain \(\{0, 2, 4, 6\}\):
- For \(x = 0\): \(y = \frac{1}{2}(0) + 1 = 0 + 1 = 1\). The ordered pair is \((0, 1)\).
- For \(x = 2\): \(y = \frac{1}{2}(2) + 1 = 1 + 1 = 2\). The ordered pair is \((2, 2)\).
- For \(x = 4\): \(y = \frac{1}{2}(4) + 1 = 2 + 1 = 3\). The ordered pair is \((4, 3)\).
- For \(x = 6\): \(y = \frac{1}{2}(6) + 1 = 3 + 1 = 4\). The ordered pair is \((6, 4)\).
- Conclusion and Final Results
Having calculated the y-values for each x-value in the domain, we have identified the following four points that satisfy the function \(y = \frac{1}{2}x + 1\):
\((0, 1)\), \((2, 2)\), \((4, 3)\), and \((6, 4)\). These points represent the relationship defined by the function within the specified domain. Plotting these points on a coordinate plane provides a visual representation of this linear relationship. The final set of ordered pairs is \(\boxed{\{(0, 1), (2, 2), (4, 3), (6, 4)\}}\).
Examples
Understanding how to calculate and plot ordered pairs is fundamental in many real-life applications. For example, if you are planning a road trip where your car travels at a constant speed, you can use a linear function to determine your distance from home over time. By knowing your speed, you can create a function like \(d = vt + d_0\) where \(d\) is distance, \(v\) is velocity, \(t\) is time, and \(d_0\) is your starting point. Plotting these points helps you estimate when you will arrive at your destination or how far you have traveled at any given time during your journey.