The first question asks us to find $g^{-1}(-8)$ given that $g(x) = x^3$ and $g(-2) = -8$. The second question asks us to interpret the meaning of $T^{-1}(258) = 35$, where $T(x)$ is the temperature of a cake in degrees Fahrenheit in an oven $x$ minutes after the cake is placed in the oven.

1. Problem Description

The first question asks us to find $g^{-1}(-8)$ given that $g(x) = x^3$ and $g(-2) = -8$.

The second question asks us to interpret the meaning of $T^{-1}(258) = 35$, where $T(x)$ is the temperature of a cake in degrees Fahrenheit in an oven $x$ minutes after the cake is placed in the oven.

2. Solution Steps

For the first question:

We are given that $g(x) = x^3$ and $g(-2) = -8$.

We want to find $g^{-1}(-8)$.

Since $g(-2) = -8$, applying the inverse function to both sides gives $g^{-1}(g(-2)) = g^{-1}(-8)$.

Since $g^{-1}(g(x)) = x$, we have $-2 = g^{-1}(-8)$.

Therefore, $g^{-1}(-8) = -2$.

For the second question:

The function $T(x)$ represents the temperature of a cake in degrees Fahrenheit after $x$ minutes in the oven.

The inverse function $T^{-1}(y)$ represents the number of minutes it takes for the cake to reach a temperature of $y$ degrees Fahrenheit.

We are given $T^{-1}(258) = 35$.

This means that it takes 35 minutes for the cake to reach a temperature of 258 degrees Fahrenheit. Therefore, after 35 minutes, the cake is 258 degrees Fahrenheit.

3. Final Answer

For the first question:

$-2$

For the second question, the correct option is:

The cake is 258 degrees Fahrenheit after 35 minutes.