Let $s$ be the side length of cube B. Write an equation that shows its surface area $6s^2$ is $6\text{ cm}^2$ greater than cube A’s surface area $6x^2$.

Once you have an equation connecting $s$ and $x$, solve for $s$ in terms of $x$. Then substitute this expression for $s$ into $s^3$ to get the volume of cube B as a function of $x$.

After you find an expression for the volume, rewrite any square roots using rational exponents (like $1/2$ or $3/2$) so you can compare it directly with the answer options.

In Desmos, type SA(x) = 6x^2 to represent the surface area of cube A as a function of $x$.

On four new lines, type the candidate functions:

• g1(x) = (x + 1)^3
• g2(x) = (x^2 + 1)^2
• g3(x) = (x + 1)^(3/2)
• g4(x) = (x^2 + 1)^(3/2) These represent possible formulas for cube B’s volume in terms of $x$.

For each candidate, create a side-length function by taking the cube root (since volume $= s^3$). For example:

• s1(x) = (g1(x))^(1/3)
• s2(x) = (g2(x))^(1/3)
• s3(x) = (g3(x))^(1/3)
• s4(x) = (g4(x))^(1/3) These represent the implied side length of cube B for each choice.

Now compute cube B’s surface area from each side-length function:

• SB1(x) = 6*(s1(x))^2
• SB2(x) = 6*(s2(x))^2
• SB3(x) = 6*(s3(x))^2
• SB4(x) = 6*(s4(x))^2 These are the surface areas that each candidate volume would imply.

For each, graph the difference between B’s and A’s surface areas:

• D1(x) = SB1(x) - SA(x)
• D2(x) = SB2(x) - SA(x)
• D3(x) = SB3(x) - SA(x)
• D4(x) = SB4(x) - SA(x) Look at the graphs and determine which $D_i(x)$ is constantly equal to $6$ (a horizontal line at $y=6$ for positive $x$). The corresponding $g_i(x)$ is the correct definition of $g$.

For any cube with side length $s$:

• Surface area: $6s^2$ (6 faces, each with area $s^2$)
• Volume: $s^3$

Cube A has side length $x$, so its surface area is $6x^2$.

Let the side length of cube B be $s$.

• Surface area of cube A: $6x^2$
• Surface area of cube B: $6\text{ cm}^2$ greater than cube A

So the surface area of cube B is $6x^2 + 6$.

But the formula for cube B’s surface area using its side $s$ is also $6s^2$.

Set these equal:

Start from

Divide both sides by $6$:

Since side length is positive, take the positive square root:

So cube B’s side length is expressed in terms of $x$ as $\sqrt{x^2 + 1}$.

The volume of a cube is $s^3$. For cube B,

Rewrite the power of a square root using a rational exponent:

• $\sqrt{x^2 + 1} = (x^2 + 1)^{1/2}$
• So $\big(\sqrt{x^2 + 1}\big)^3 = \big((x^2 + 1)^{1/2}\big)^3 = (x^2 + 1)^{3/2}$

Therefore, the function $g$ that gives the volume of cube B in terms of $x$ is