Theory of Intelligence • First-Principles AI & STEM Education

Step-by-Step Solution

Step 1: Identify Range of $f(x)$

The parabola has its vertex at $(-4, -5)$ and opens downwards ( $a = -3 < 0$ ). For the restricted domain $x < -4$ :

x + 4 < 0 \implies (x + 4)^2 > 0 \implies -3(x + 4)^2 < 0 \implies -3(x + 4)^2 - 5 < -5

As $x \to -\infty$ , $f(x) \to -\infty$ . Thus, the range of $f$ is:

\text{Ran}(f) = (-\infty, -5)

Step 2: Solve for $x$ in Terms of $y$

Set $y = -3(x + 4)^2 - 5$ :

\begin{aligned} y + 5 &= -3(x + 4)^2 \\ (x + 4)^2 &= -\frac{y + 5}{3} \end{aligned}

Note: Since $y \in (-\infty, -5)$ , $y < -5 \implies y + 5 < 0$ , which makes $-\frac{y + 5}{3} > 0$ . The radicand is guaranteed to be strictly positive!

Taking the square root:

x + 4 = \pm\sqrt{-\frac{y + 5}{3}} \implies x = -4 \pm\sqrt{-\frac{y + 5}{3}}

Step 3: Justify the Selection of the Radical Sign ( $\pm$ )

Because $x < -4$ , we have $x + 4 < 0$ .

The positive root $-4 + \sqrt{-\frac{y+5}{3}} \ge -4$ violates the domain requirement $x < -4$ .
Therefore, we must choose the negative root:

x = -4 - \sqrt{-\frac{y + 5}{3}}

Step 4: Swap Variables and State Domains

Interchanging variables gives the inverse formula:

f^{-1}(x) = -4 - \sqrt{-\frac{x + 5}{3}}

Stating the swapped intervals:

\begin{aligned} \text{Dom}(f^{-1}) &= \text{Ran}(f) = (-\infty, -5) \\ \text{Ran}(f^{-1}) &= \text{Dom}(f) = (-\infty, -4) \end{aligned}

Step-by-Step Solution

Step 1: Identify Range of f(x)f(x)f(x)

Step 2: Solve for xxx in Terms of yyy

Step 3: Justify the Selection of the Radical Sign (±\pm±)

Step 4: Swap Variables and State Domains

Discussion & Community

Step 1: Identify Range of $f(x)$

Step 2: Solve for $x$ in Terms of $y$

Step 3: Justify the Selection of the Radical Sign ( $\pm$ )