Theory of Intelligence • First-Principles AI & STEM Education

Step-by-Step Solution

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

The quadratic is already presented in vertex form:

f(x) = 2(x - 3)^2 + 1

The vertex is $(3, 1)$ . For $x \ge 3$ , $(x - 3) \ge 0 \implies 2(x - 3)^2 \ge 0$ . The minimum output value is $f(3) = 1$ . As $x \to \infty$ , $f(x) \to \infty$ . Therefore:

\text{Ran}(f) = [1, \infty)

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

Set $y = 2(x - 3)^2 + 1$ :

\begin{aligned} y - 1 &= 2(x - 3)^2 \\ (x - 3)^2 &= \frac{y - 1}{2} \\ x - 3 &= \pm\sqrt{\frac{y - 1}{2}} \\ x &= 3 \pm\sqrt{\frac{y - 1}{2}} \end{aligned}

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

Because the domain is restricted to $x \ge 3$ , the value $x - 3$ is non-negative:

x - 3 \ge 0 \implies x = 3 + \sqrt{\frac{y - 1}{2}}

The negative branch $3 - \sqrt{\frac{y-1}{2}}$ would produce values $\le 3$ , which are outside the specified domain.

Step 4: Swap Variables and State Domains

Interchanging variables $x$ and $y$ :

f^{-1}(x) = 3 + \sqrt{\frac{x - 1}{2}}

The domain and range of the inverse are:

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