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 \( f(x) = x^3 + 1 \) across the x-axis, you need to negate the output of the function.

The reflection across the x-axis would mean that if \( f(x) = y \), then the reflected graph would be \( g(x) = -y \).

So, apply this to the equation:

1. Start with the original function: \[ f(x) = x^3 + 1 \]

2. Reflect it across the x-axis: \[ g(x) = -(x^3 + 1) = -x^3 - 1 \]

Thus, the equation of the dotted graph is \( g(x) = -x^3 - 1 \).

Therefore, the answer is Option #2.