Show a function is monotonic and find a formala for its inverse

Let $f(x) = x^3$ . Show that $f$ is strictly monotonic on $\mathbb{R}$ . Find the domain of the inverse of $f$ , denoted by $g$ . Find a formula for computing $g(y)$ for each $y$ in the domain of $g$ .

First, to show $f$ is monotonic let $x_1, x_2 \in \mathbb{R}$ with $x_1 < x_2$ . Then we consider

$\begin{align*} x_2^3 - x_1^3 &= (x_2 - x_1)(x_2^2 + x_1x_2 + x_1^2)\\ &= (x_2 - x_1)\left( \left( x_2 + \frac{x_1}{2} \right)^2 + \frac{3x_1^2}{4} \right). \end{align*}$

But, since $x_2 > x_1$ by assumption, we have $(x_2 - x_1) > 0$ . The second term in the product is also positive since it is a sum of positive terms. Therefore,

$x_2^3 - x_1^3 > 0 \qquad \implies \qquad f(x_2) > f(x_1).$

Hence, $f$ is strictly increasing on $\mathbb{R}$ . (Of course, there are faster ways to discover the $f(x) =x^3$ is increasing, but we do not know yet what is a derivative.)

Next,

$y = x^3 \quad \implies \quad x = y^{\frac{1}{3}} \quad \implies \quad g(y) = y^{\frac{1}{3}} \quad \text{for all } y \in \mathbb{R}.$

Stumbling Robot

Show a function is monotonic and find a formala for its inverse

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