Question

question 24

the point \\((-1, 1)\\) is on the graph of \\(y = f(x)\\). determine the new coordinates of this point on the graph of \\(g(x) = -2f\left(\frac{1}{3}x + 1\
ight) - 3\\).

• \\((-6, -5)\\)
• \\((6, 5)\\)
• not enough information has been provided.
• none of these are correct points.
• \\((-6, 5)\\)
• \\((6, -5)\\)

Explanation:

⚡ Using what you learned: combining transformations

Step 1: Find the new x-coordinate

Set the input of the transformed function equal to the original x-coordinate:

$$ \frac{1}{3}x + 1 = -1 $$

Solve for \( x \):

$$ \frac{1}{3}x = -2 $$

$$ x = -6 $$

Step 2: Find the new y-coordinate

Substitute the original output \( f(-1) = 1 \) into the outer transformations:

$$ y = -2 \cdot f(-1) - 3 $$

$$ y = -2(1) - 3 $$

$$ y = -5 $$

Answer:

\( (-6, -5) \)