Let's break down the problem given in the image for the function $f(x) = -2x + 1$:

a. Sketch the graph of the function

To sketch the graph of $f(x) = -2x + 1$:

• The equation is in slope-intercept form $y = mx + b$, where $m = -2$ is the slope and $b = 1$ is the y-intercept.
• This means the line crosses the y-axis at $(0, 1)$ and has a slope of -2, meaning for every 1 unit you move to the right, the graph moves down by 2 units.

b. Find the slope and y-intercept

• Slope (m): The slope is $m = -2$.
• Y-intercept (b): The y-intercept is $(0, 1)$.

c. Find its inverse

To find the inverse of the function:

1. Start with $y = -2x + 1$.
2. Swap $x$ and $y$ to find the inverse: $x = -2y + 1$
3. Solve for $y$: $x - 1 = -2y \implies y = \frac{1 - x}{2}$ Thus, the inverse function is: $f^{-1}(x) = \frac{1 - x}{2}$

d. Sketch the graph of its inverse

The graph of the inverse $f^{-1}(x) = \frac{1 - x}{2}$ is a line with:

• Slope $m = -\frac{1}{2}$,
• Y-intercept $b = \frac{1}{2}$.

This line crosses the y-axis at $(0, 0.5)$ and slopes downward with a gentler slope than the original function.

Do you want further details or assistance on any of these parts?

Here are 5 related questions to expand the understanding:

1. How does the slope of a line affect the steepness and direction of the graph?
2. What does it mean when the slope of a function is negative?
3. How do you determine the domain and range of the inverse function?
4. Can a function and its inverse intersect? If yes, where do they intersect?
5. How does switching the x and y coordinates visually change the graph when finding the inverse?

Tip: When graphing a function and its inverse, they will always be reflections of each other across the line $y = x$.