https://file.aiquickdraw.com/mathbot/file/1727082724224e9ud0a19.jpg

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.

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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 \( y = x \).

For the function f(x) = -2x + 1: a) Sketch the graph of the function. b) Find the slope and y-intercept. c) Find its inverse. d) Sketch the graph of its inverse.

Learn how to sketch the graph of the linear function f(x) = -2x + 1, find its slope and y-intercept, calculate its inverse, and sketch the graph of the inverse. Suitable for students in Grades 9-12, this problem covers linear equations and inverse functions.

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