The problem involves rotating the linear function $f(x) = -2x + 3$ counterclockwise, and we need to determine the possible equation of its transformed function $g(x)$.

Key Insight:

When a function is rotated counterclockwise around the origin by $90^\circ$, the coordinates $(x, y)$ of the points on the graph are transformed into $(-y, x)$. In terms of equations, the slope $m$ of the line changes to $-1/m$, and the y-intercept may change based on the context.

The given function is $f(x) = -2x + 3$. Its slope is $-2$. After a $90^\circ$ counterclockwise rotation:

1. The new slope becomes $\frac{-1}{-2} = \frac{1}{2}$.
2. The transformation modifies the line, resulting in a new equation.

Analyzing the Options:

We test each provided $g(x)$:

1. $g(x) = x - 3$: Slope is $1$, not $\frac{1}{2}$. Incorrect.
2. $g(x) = -4x + 3$: Slope is $-4$, not $\frac{1}{2}$. Incorrect.
3. $g(x) = -x + 3$: Slope is $-1$, not $\frac{1}{2}$. Incorrect.
4. $g(x) = 3x - 2$: Slope is $3$, not $\frac{1}{2}$. Incorrect.

Rechecking:

Based on the provided options, none explicitly match the expected rotated equation. It seems there may be an error in the answer choices or assumptions. If you have clarifications or additional details, I can refine the explanation further.

Would you like me to detail the rotation process further or explore transformations in more depth?

Related Questions:

1. What is the general process for rotating a linear function by $90^\circ$?
2. How does the slope of a line transform during a counterclockwise rotation?
3. Why does the y-intercept shift when rotating a function about the origin?
4. How can transformations be visualized geometrically on a Cartesian plane?
5. What happens to nonlinear functions during rotations?

Tip:

When rotating functions, always transform the slope first, then adjust the intercept by substituting known points if needed.