Question

Use the image to answer the question.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 \( f(x) = x^3 + 1 \), we need to reflect it across the x-axis.

Reflecting a function \( f(x) \) across the x-axis results in a new function \( g(x) = -f(x) \).

Starting with the given function:
\[
f(x) = x^3 + 1
\]

The reflection will be:
\[
g(x) = -f(x) = -(x^3 + 1) = -x^3 - 1
\]

Now, we can compare this with the options provided:

- Option #1: \( g(x) = -x^2 + 1 \)
- Option #2: \( g(x) = -x^3 - 1 \)
- Option #3: \( g(x) = -x^3 + 1 \)
- Option #4: \( g(x) = x^3 - 1 \)

The correct answer is:

**Option #2: \( g(x) = -x^3 - 1 \)**