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The Lumenary · Grades 6–8

Function machines in sequence

Name: ______________________

Date: ______________________

You will learn to

Compute the result of applying one function and then another, using tables, arrow diagrams and formulas
Decide whether a chain is possible, by checking that every output of the first function is an allowed input of the second
Recognise the identity function and explain why chaining it before or after any function changes nothing
Find the inverses of simple bijections such as $x \mapsto x + 3$ and $x \mapsto 2x$ on $\Q$ , and undo a chain by undoing its steps in reverse order
Explain why a function that sends two inputs to the same output, such as $x \mapsto x^2$ on $\Z$ , cannot be undone

1.
Let $f(x) = 3x - 1$ and $g(x) = x^2$ . Apply $f$ , then $g$ , to the input $2$ . What is the output?
2.
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.
3.
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.“square, then take the square root”
- B.“subtract $1$ , then take the reciprocal $\tfrac{1}{x}$ ”
- C.“add $1$ , then take the square root”
- D.“square and change the sign ( $x \mapsto -x^2$ ), then take the square root”
4.
A function is the chain “multiply by $4$ , then subtract $7$ ”. Which chain undoes it?
- A.add $7$ , then divide by $4$
- B.divide by $4$ , then add $7$
- C.subtract $7$ , then divide by $4$
- D.add $7$ , then multiply by $4$
5.
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?
6.
Which of these functions from $\Z$ to $\Z$ can be undone by some function from $\Z$ to $\Z$ ? Choose all that apply.
- A. $x \mapsto x + 10$
- B. $x \mapsto 3x$
- C. $x \mapsto x^2$
- D. $x \mapsto |x|$ (the distance from $0$ )
- E. $x \mapsto -x$
7.
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?
8.
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?
- A.Exactly when $g(f(a)) = f(g(a))$ for every $a$ — that is, when the order of $f$ and $g$ does not matter
- B.Only when $f = g$
- C.Only when $f$ or $g$ is the identity
- D.Never
9.
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.¬ ∃ g : Int → Int, Undoes g (fun x => x * x)
- B.∃ g : Int → Int, ¬ Undoes g (fun x => x * x)
- C.¬ ∃ g : Int → Int, Undoes (fun x => x * x) g
- D.∀ g : Int → Int, Undoes g (fun x => x * x)
10.
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).
Computational — Practice

Answer key — Function machines in sequence

1.
$f(2) = 3 \cdot 2 - 1 = 5$ , and then $g(5) = 5^2 = 25$ . In symbols, $g(f(2)) = 25$ . (The other order gives $f(g(2)) = f(4) = 11$ .)
2.
$1 \mapsto a \mapsto y$ , $2 \mapsto c \mapsto y$ , $3 \mapsto a \mapsto y$ and $4 \mapsto b \mapsto x$ . So the inputs sent to $y$ are $\{1, 2, 3\}$ .
3.
Only “square, then take the square root” works for every integer, because $x^2 \ge 0$ is always an allowed input of $\sqrt{\ }$ . Each other chain breaks somewhere: at $x = 1$ , at $x = -5$ (or any $x < -1$ ), and at every $x \neq 0$ respectively.
4.
By the socks-and-shoes rule (Theorem 3), undo “subtract $7$ ” first (add $7$ ), then undo “multiply by $4$ ” (divide by $4$ ): $x \mapsto 4x - 7 \mapsto 4x \mapsto x$ . Test with $x = 3$ : $3 \mapsto 5$ , and $5 + 7 = 12$ , $12 \div 4 = 3$ .
5.
Undo the steps from last to first:
- $2 \times 10 = 20$ (undo “divide by $10$ ”);
- $20 - 8 = 12$ (undo “add $8$ ”);
- $12^2 = 144$ (undo “square root”);
- $144 + 52 = 196$ (undo “subtract $52$ ”);
- $\sqrt{196} = 14$ (undo “multiply by itself”);
- $14 \times \tfrac{3}{2} = 21$ (undo “subtract a third”, which multiplied by $\tfrac{2}{3}$ );
- $21 \times 7 = 147$ (undo “divide by $7$ ”);
- $147 \times \tfrac{4}{7} = 84$ (undo “add three quarters”, which multiplied by $\tfrac{7}{4}$ );
- $84 \div 3 = 28$ (undo “multiply by $3$ ”).
Check forwards: $28 \to 84 \to 147 \to 21 \to 14 \to 196 \to 144 \to 12 \to 20 \to 2$ .
6.
$x \mapsto x + 10$ , $x \mapsto 3x$ and $x \mapsto -x$ can be undone (by subtracting $10$ , by dividing by $3$ and rounding down, and by $x \mapsto -x$ itself). $x \mapsto x^2$ and $x \mapsto |x|$ cannot, because each sends two different inputs ( $\pm 2$ , respectively $\pm 3$ ) to the same output.
7.
The chains give $5x + 5a$ and $5x + a$ . These agree for every $x$ exactly when $5a = a$ , i.e. $4a = 0$ , i.e. $a = 0$ . This is Theorem 2 with $m = 5$ .
8.
Suppose $g^{-1}(f^{-1}(g(f(a)))) = a$ . Applying $g$ to both sides gives $f^{-1}(g(f(a))) = g(a)$ , and applying $f$ gives $g(f(a)) = f(g(a))$ . Conversely, if $g(f(a)) = f(g(a))$ for all $a$ , then $g^{-1}(f^{-1}(g(f(a)))) = g^{-1}(f^{-1}(f(g(a)))) = g^{-1}(g(a)) = a$ . So the wrong order works exactly when $f$ and $g$ can be swapped — as the lesson's experiment found for all $36$ pairs on a three-element set.
9.
“Cannot be undone” is “there is no $g$ with $g$ undoing squaring”: ¬ ∃ g : Int → Int, Undoes g (fun x => x * x). Moving the ¬ inside (∃ g, ¬ …) gives a much weaker statement, as the quantifiers lesson explains.
10.
```
def invert(table):
    g = {}
    for a, b in table.items():
        if b in g:
            return None
        g[b] = a
    return g
```
If two inputs share an output, the function cannot be undone (Theorem 4), and the function returns None. Otherwise every output has exactly one input, and g sends it back. Notice that g is defined only on the outputs of table: undoing only needs to work on outputs.