EC-18.1 Functions, and the Language of Change

What this is and why it exists

Every subject downstream of this one opens by writing a function and assuming you can read it. A circuits course writes a voltage as a function of time. A devices course writes a current as a function of an applied voltage. A communications course writes a signal as a function of frequency.

None of them stops to say what that notation means. Each assumes a school course did it, and for many readers no course did.

So this topic does it once, on its own. What is here is not calculus yet. It is the object calculus acts on, together with the handful of families that object nearly always belongs to.

The vocabulary

• Function — a rule that gives exactly one output for each admissible input.
• Domain — the set of inputs the rule admits. This is part of the function, not an afterthought.
• Range — the set of outputs the rule actually produces.
• Independent variable — the input, the quantity you choose.
• Dependent variable — the output, the quantity that follows.
• Interval — a connected piece of the real line. It is open when its ends are excluded and closed when they are included.
• Composition — feeding the output of one function into another.
• Inverse — the rule that undoes a function, in the cases where one exists.
• One-to-one — never sending two different inputs to the same output.
• Model — a function chosen to stand for a physical relationship.

The mental model

Start with the part that is most often skipped. A function is a rule together with the set of inputs it admits. Two rules that agree wherever both are defined, but disagree about what is admissible, are different functions.

That sounds like bookkeeping. It is not. In engineering the domain is where the physical situation makes sense. A model of a heated resistor holds for a range of temperatures and says nothing outside it. Losing the domain loses the boundary of what the model asserts about the world.

Next, the families. Almost every model in this area is built from six shapes, and it is worth being able to recognise each on sight.

• Linear — a constant rate of change. A straight line. Doubling the input doubles the change in the output.
• Polynomial — sums of whole-number powers. Smooth, with a limited number of turns.
• Rational — one polynomial divided by another. These blow up where the lower polynomial is zero, and that is where transfer functions get their poles.
• Exponential — a fixed multiplying factor per unit of input. This is what every decaying transient in a circuit does.
• Logarithmic — the inverse of the exponential. This is why decibels exist.
• Trigonometric — repeating with a fixed period. Every alternating quantity in this area is one of these or a sum of them.

Choosing which family a measurement resembles is most of the work of modelling. Plotting the data and asking which shape it looks like is a legitimate first step, not a shortcut.

Now composition and inverses. Physical effects come in chains. Temperature moves a sensor voltage, the sensor voltage moves an amplifier output, the amplifier output moves a reading. Writing that chain down is composition, and it is why the chain rule later is the rule you use most.

Undoing a chain needs an inverse at each step. An inverse exists only when the original rule is one-to-one. This is a real restriction. Squaring is not one-to-one over all the real numbers, because two inputs of opposite sign give the same output. Restricting the domain to the non-negative numbers makes it one-to-one, and the square root is then its inverse. Restricting the domain to obtain an inverse is a decision, and the decision has to be recorded.

Finally, units. A function that models something physical has units on both sides. An expression whose units do not balance is wrong before any number is computed. Checking units is the cheapest error detector available in this subject, and it costs seconds.

What you should now be able to explain or do

• State the domain and range of a function, and say why the domain is part of the function.
• Recognise the six common families from a graph and say what each one does physically.
• Write a chain of physical effects as a composition of functions.
• Decide whether a function has an inverse, and restrict its domain until it does.
• Read a graph for where a quantity increases, decreases, turns and stops being defined.
• Check that an expression balances in units before computing anything with it.

Check yourself