Let's break down the problem step by step.

The function provided is:

$g(x) = -2x + 1$

You are tasked with finding:

1. (a) The domain and range of $g(x)$

   • Domain: The domain of a linear function like $g(x) = -2x + 1$ is all real numbers, i.e., $(-\infty, \infty)$.
   • Range: Since the function is linear and not restricted, the range is also all real numbers, $(-\infty, \infty)$.

2. (b) Evaluate $g(-2)$ and $g(1)$ using the formula:

   • $g(x) = -2x + 1$
     • $g(-2) = -2(-2) + 1 = 4 + 1 = 5$
     • $g(1) = -2(1) + 1 = -2 + 1 = -1$

3. (c) Use the graph of $g(x)$ to evaluate $g(-2)$ and $g(1)$:
   By checking the graph at $x = -2$ and $x = 1$, you should see that:

   • At $x = -2$, $g(-2) = 5$
   • At $x = 1$, $g(1) = -1$

Thus, the graph confirms the values obtained using the formula.

Do you want further details or have any questions about this process?

Here are 5 related questions to deepen your understanding:

1. What is the slope of the function $g(x) = -2x + 1$, and how does it affect the graph?
2. How would the domain and range change if the function had been a quadratic instead of linear?
3. What other methods can be used to evaluate functions at specific points aside from using the graph and formula?
4. How does the sign of the slope ($-2$) influence the direction of the line?
5. How can you check the accuracy of your graph by evaluating points on the line?

Tip: For linear functions, the domain is always all real numbers unless explicitly restricted.