Theory of Intelligence • First-Principles AI & STEM Education

Step-by-Step Solution

Step 1: Analyze Behavior on the Domain & Determine Range

The vertex of the full quadratic occurs at:

h = -\frac{b}{2a} = -\frac{-2}{2(3)} = \frac{1}{3}

Because the restricted domain is $x \ge 5$ , which lies entirely to the right of the vertex $x = \frac{1}{3}$ , the function $f(x)$ is strictly increasing on $[5, \infty)$ .

The minimum value on this interval occurs at the left boundary $x = 5$ :

f(5) = 3(5)^2 - 2(5) + 6 = 3(25) - 10 + 6 = 75 - 4 = 71

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

\text{Ran}(f) = [71, \infty)

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

Set $y = 3x^2 - 2x + 6$ , yielding the quadratic equation in $x$ :

3x^2 - 2x + (6 - y) = 0

Applying the quadratic formula with $A = 3$ , $B = -2$ , $C = 6 - y$ :

\begin{aligned} x &= \frac{-(-2) \pm\sqrt{(-2)^2 - 4(3)(6 - y)}}{2(3)} \\ &= \frac{2 \pm\sqrt{4 - 72 + 12y}}{6} \\ &= \frac{2 \pm\sqrt{12y - 68}}{6} \\ &= \frac{2 \pm\sqrt{4(3y - 17)}}{6} \\ &= \frac{2 \pm 2\sqrt{3y - 17}}{6} \\ &= \frac{1 \pm\sqrt{3y - 17}}{3} \end{aligned}

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

At the boundary value $y = 71$ :

For the positive sign: $x = \frac{1 + \sqrt{3(71) - 17}}{3} = \frac{1 + \sqrt{213 - 17}}{3} = \frac{1 + \sqrt{196}}{3} = \frac{1 + 14}{3} = \frac{15}{3} = 5$ . This satisfies $x \ge 5$ .
For the negative sign: $x = \frac{1 - 14}{3} = -\frac{13}{3} \approx -4.33 < 5$ , violating the domain condition.

Hence, we must select the positive root:

x = \frac{1 + \sqrt{3y - 17}}{3}

Step 4: Swap Variables and State Domains

Interchanging variables:

f^{-1}(x) = \frac{1 + \sqrt{3x - 17}}{3}

The swapped intervals are:

\begin{aligned} \text{Dom}(f^{-1}) &= \text{Ran}(f) = [71, \infty) \\ \text{Ran}(f^{-1}) &= \text{Dom}(f) = [5, \infty) \end{aligned}

Step-by-Step Solution

Step 1: Analyze Behavior on the Domain & Determine Range

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

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