The Observatory's Inside-Out Signal

At a glance

• 1. Δ(𝑡) =𝑡 −3
• 2. 𝑅(𝑡) =𝑆(Δ(𝑡)) =𝑆(𝑡 −3)
• 3. 𝐿(𝑥) =2𝑥 +5
• 4. 𝐴(𝑡) =𝐿(𝑅(𝑡)) =2𝑆(𝑡 −3) +5
• 5. 𝐸(𝑡) =𝐶(𝐴(𝑡))
• 6. 𝐸(𝑡) =𝐶(𝐿(𝑆(𝑡 −3)))
• 7. 𝐸 =𝐶 ∘𝐿 ∘𝑆 ∘Δ, with Δ acting first.
• 8. 𝐿−1(𝑦) =𝑦−52
• 9. 𝑆(𝑡) =𝐿−1(𝐶−1(𝐸(𝑡 +3))) =𝐶−1(𝐸(𝑡+3))−52

1. Define the delay map

The delay map sends the displayed time to the raw-signal time three seconds earlier:

2. Apply the delay to the raw signal

Applying the raw-signal function after that map gives

3. Define the lift lens

The lift lens doubles an input and adds 5, so

4. Apply the lift to the delayed signal

Its input is the delayed raw signal. Therefore

5. Apply the calibration lens

The calibration lens acts on the lifted signal:

6. Nest the complete forward rule

Substituting the earlier stages creates the fully nested equation

7. Write the composition equation

As a composition of named functions, the instrument is

Composition is read from right to left when it acts: Δ changes the time first, then 𝑆 returns the delayed raw reading, then 𝐿 lifts it, and finally 𝐶 calibrates it.

8. Invert and verify the lift lens

Start with 𝑦 =2𝑥 +5 and solve for 𝑥:

Thus

The requested verification is

9. Recover the current raw signal from the future display

Evaluate the forward equation three seconds in the future:

The last forward stage was 𝐶, so the first undoing stage must be 𝐶−1. The lift lens was immediately before it, so 𝐿−1 acts next:

Substituting the formula for 𝐿−1 gives

Check

Send the recovered expression forward through 𝐿:

Then apply 𝐶:

The recovery formula returns the supplied future display, so the inverse order is consistent.