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Show that a function is monotonic and find a formula for its inverse

by RoRi

Let

    \[ f(x) = \begin{cases} x & \text{if } x < 1, \\ x^2 & \text{if } 1 \leq x \leq 4, \\ 8x^{\frac{1}{2}} & \text{if } x > 4. \end{cases} \]

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 strictly increasing on \mathbb{R} we note that it is strictly increasing on each component (since x^2, \ x, and 8x^{1/2} are all increasing functions on the domains given). Then we must consider the transition from one of these regions to another. (In other words, we know the function is increasing on each interval, but we need to check that it is increasing from one interval to the next.)

    \[ x_1 < 1 \leq x_2 \quad \implies \quad x_1 < x_2^2 \quad \implies \quad f(x_1) < f(x_2), \]

and,

    \[ 1 \leq x_1 \leq 4 < x_2 \quad \implies \quad x_1^2 < 8x_2^{\frac{1}{2}} \quad \implies \quad f(x_1) < f(x_2). \]

Thus, f is indeed increasing on all of \mathbb{R}.
Next,

    \[ g(y) = \begin{cases} y & \text{if } y < 1, \\ y^{\frac{1}{2}} & \text{if } 1 \leq y \leq 16, \\ \left( \frac{y}{8}\right)^2 & \text{if } y > 16. \end{cases} \]

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