We are given that $(2, 1)$ is a point on the graph of $y = f(x)$, which means:

$f(2) = 1$

We need to find the corresponding point on the graph of the function $y = 1 - f(1 - x)$.

Step-by-step transformation:

1. Input transformation: In the function $y = 1 - f(1 - x)$, the $x$-coordinate is replaced by $1 - x$. This represents a horizontal reflection about the vertical line $x = 1$ (flipping across $x = 1$).

2. Output transformation: The function value is then modified by subtracting it from 1 (this represents a vertical reflection and a vertical shift).

We now apply this to the point $(2, 1)$:

• The $x$-coordinate becomes $1 - x = 1 - 2 = -1$.
• The function value at $x = 2$ is $f(2) = 1$, so now we calculate: $y = 1 - f(1 - 2) = 1 - f(-1)$ Since we don't know the specific value of $f(-1)$, we can't compute it directly unless we have information about the function $f$. However, the transformed point is $(-1, 1 - f(-1))$.

So the corresponding point on the graph of $y = 1 - f(1 - x)$ is:

$\boxed{(-1, 1 - f(-1))}$

Would you like to discuss any further steps or clarifications?

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5 Related Questions:

1. How do transformations of functions like reflections and shifts affect the graph of the function?
2. What does it mean for a function to be reflected across a specific line?
3. How can we determine the transformation of points when the function involves a combination of shifts and reflections?
4. What does $f(x) = 1 - f(1 - x)$ represent in terms of transformations?
5. Can you identify the properties of the function $f(x)$ from the given point?

Tip:

Remember that a reflection across the line $x = a$ changes the input by replacing $x$ with $2a - x$.