Let's solve the given problem. Here's the situation:

We are asked to determine the transformed equation $g(x)$ after rotating the function $f(x) = -2x + 3$ counterclockwise.

Step 1: Analyze the rotation.

When a function is rotated counterclockwise about the origin by $90^\circ$, the transformation involves swapping the $x$- and $y$-coordinates and applying a sign change. For the line $f(x) = y = -2x + 3$, the steps are:

1. Let $x$ and $y$ represent the original variables: $y = -2x + 3$.
2. After rotation by $90^\circ$, the new coordinates become: $x' = -y, \quad y' = x.$

Step 2: Substitute and rearrange.

The rotated relationship becomes: x = -(-2x + 3) \quad \text{(substitute y = -2x + 3$into$x' = -y)}.

Simplify: $x = 2x - 3.$

Rearrange for $y'$ (which is the new $g(x)$): $y' = x - 3.$

Thus, the transformed equation is: $g(x) = x - 3.$

Step 3: Match with the options.

From the given options, the correct answer is: $g(x) = x - 3.$

Final Answer:

The transformed function is $g(x) = x - 3$.

Would you like more details or explanations? Let me know!

5 Related Questions:

1. How does rotation of a function affect its slope?
2. What is the transformation rule for rotating a line by $90^\circ$ clockwise?
3. How would the function transform if it were reflected instead of rotated?
4. What happens to the y-intercept of a function after rotation?
5. Can this concept apply to non-linear functions, such as parabolas?

Tip:

When rotating functions, always consider the geometric effect on the slope and intercept. It's helpful to plot the function to visualize the transformation.