Function machines in sequence

Let $f(x) = 3x - 1$ and $g(x) = x^2$. Apply $f$, then $g$, to the input $2$. What is the output?

Let $f : \{1, 2, 3, 4\} \to \{a, b, c\}$ be given by $f(1) = a$, $f(2) = c$, $f(3) = a$, $f(4) = b$, and $g : \{a, b, c\} \to \{x, y\}$ by $g(a) = y$, $g(b) = x$, $g(c) = y$. Which inputs does “$f$, then $g$” send to $y$? List them.

Which of these chains is defined for every integer input $x$? (Here $\sqrt{x}$ accepts only inputs $x \ge 0$, and $\tfrac{1}{x}$ only inputs $x \neq 0$.)

A function is the chain “multiply by $4$, then subtract $7$”. Which chain undoes it?

From Bhāskara II's Līlāvatī (1150): a number is multiplied by $3$; then three quarters of the result is added to it; the sum is divided by $7$; a third of the quotient is subtracted from it; the result is multiplied by itself; $52$ is subtracted; the square root is taken; $8$ is added; and the result is divided by $10$. The answer is $2$. What was the number?

Which of these functions from $\Z$ to $\Z$ can be undone by some function from $\Z$ to $\Z$? Choose all that apply.

For which integer $a$ do “add $a$, then multiply by $5$” and “multiply by $5$, then add $a$” give the same output for every integer input?

Let $f$ and $g$ be functions from a set to itself with inverse functions $f^{-1}$ and $g^{-1}$. Undoing “$f$, then $g$” in the wrong order means applying $f^{-1}$, then $g^{-1}$, so each input $a$ goes to $g^{-1}\big(f^{-1}(g(f(a)))\big)$. When does this give back $a$ for every input?

In the Lean section, Undoes g f means $g(f(a)) = a$ for every $a$. Which Lean statement says “squaring on $\Z$ cannot be undone”?

A function on a finite set can be stored as a dictionary, such as {1: 'a', 2: 'b'}. Write invert(table) that returns a dictionary undoing table — sending each output back to its input — or None if two inputs share an output (so the function cannot be undone).