Find the inverse function of g(x) = 2x2 − 1. Also, determine if 2, 3, 8, and 10 are elements of set A = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12. Lastly, solve 2ⁿ = 4^(n-3).

Step 1: Replace $g(x)$ with $y$.

$$y = 2x^2 - 1$$

Step 2: Swap $x$ and $y$.

$$x = 2y^2 - 1$$

Step 3: Solve for $y$.
Add 1 to both sides:

$$x + 1 = 2y^2$$

Divide by 2:

$$\frac{x + 1}{2} = y^2$$

Take the square root of both sides. Since the original function $g(x) = 2x^2 - 1$ is not one-to-one over its entire domain, we typically restrict the domain of $g(x)$ to $x \ge 0$ (or $x \le 0$) to find an inverse function. Assuming $x \ge 0$, then $y \ge 0$ for the inverse function.

$$y = \sqrt{\frac{x + 1}{2}}$$

Step 4: Replace $y$ with $g^{-1}(x)$.

$$g^{-1}(x) = \sqrt{\frac{x + 1}{2}}$$

The domain of $g^{-1}(x)$ is $x \ge -1$.

The inverse function is:

$$\boxed{g^{-1}(x) = \sqrt{\frac{x + 1}{2}}}$$

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