The problem displayed in the image asks you to find the natural domain and graph the function $f(x) = 4 - 4x$.

Solution:

1. Natural Domain: The function $f(x) = 4 - 4x$ is a linear function. Linear functions are defined for all real numbers because there are no restrictions like division by zero or square roots of negative numbers.

   Domain: The domain is $\mathbb{R}$, which in interval notation is $(-\infty, \infty)$.

2. Graphing the Function: The function $f(x) = 4 - 4x$ can be rewritten as $f(x) = -4x + 4$, which is in the slope-intercept form $y = mx + b$, where:

   • The slope $m = -4$
   • The y-intercept $b = 4$

   To graph the function:

   • Start at the point $(0, 4)$ on the y-axis.
   • Use the slope $-4$ to find another point: go down 4 units and 1 unit to the right to reach the point $(1, 0)$.
   • Draw a straight line through these points.

Would you like a detailed explanation of how to graph the function or have any other questions?

Related Questions:

1. What are the steps to find the domain of a more complex function, like one involving square roots or fractions?
2. How does the slope of a linear function affect its graph?
3. Can you explain how to graph a function given in standard form instead of slope-intercept form?
4. What are the key characteristics of a linear function?
5. How can you find the range of a linear function?

Tip:

When determining the domain of a function, always look out for restrictions like division by zero or taking square roots of negative numbers, as these often limit the domain to a subset of real numbers.