Use the image to answer the question.

Question

Two intersecting curves are graphed on a coordinate plane.

The dotted graph is a reflection of the solid graph. The solid graph is a cubic function with the equation f(x)=x3+1 . Enter the option number that shows the equation of the image.

Option #1: g(x)=−x2+1

Option #2: g(x)=−x3−1

Option #3: g(x)=−x3+1

Option #4: g(x)=x3−1

(1 point)
The equation of the dotted graph is Option #.

Answer 1

To find the equation of the dotted graph, which is a reflection of the solid graph (the cubic function \( f(x) = x^3 + 1 \)), we need to determine the reflection across the x-axis.

The reflection of a function \( f(x) \) across the x-axis is given by \( g(x) = -f(x) \).

So, applying this to the given function:

1. Start with \( f(x) = x^3 + 1 \).
2. To reflect it, take the negative: \[ g(x) = -f(x) = -(x^3 + 1) = -x^3 - 1 \]

This matches with Option #2: \( g(x) = -x^3 - 1 \).

Therefore, the answer is:

Option #2.