Theory of Intelligence • First-Principles AI & STEM Education

Step-by-Step Solution

Step 1: Determine the Range of $f(x)$

Given the restricted domain $\text{Dom}(f) = (-\infty, 0]$ :

x \le 0 \implies x^2 \ge 0 \implies x^2 - 3 \ge -3

As $x \to -\infty$ , $x^2 \to \infty$ . Therefore, the range of $f(x)$ is:

\text{Ran}(f) = [-3, \infty)

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

Set $y = f(x)$ and isolate $x$ :

\begin{aligned} y &= x^2 - 3 \\ x^2 &= y + 3 \\ x &= \pm\sqrt{y + 3} \end{aligned}

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

Because the domain of $f$ is explicitly restricted to $x \le 0$ , $x$ cannot take positive values. Hence, we must reject the positive branch and choose the negative square root:

x = -\sqrt{y + 3}

Step 4: Swap Variables and State the Inverse Function

Interchanging $x$ and $y$ gives the inverse formula:

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

By the fundamental property of inverse functions:

\begin{aligned} \text{Dom}(f^{-1}) &= \text{Ran}(f) = [-3, \infty) \\ \text{Ran}(f^{-1}) &= \text{Dom}(f) = (-\infty, 0] \end{aligned}

Step-by-Step Solution

Step 1: Determine the 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 the Inverse Function

Discussion & Community

Step 1: Determine the Range of $f(x)$

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

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