Functions of one real variable
A function is a rule that assigns exactly one output to each input. With that idea you can describe the characteristic of a diode, the gain of an amplifier and any signal in time.
01One input, exactly one output
A function of one real variable is a rule that assigns to each number in a starting set one and only one number in a target set. That uniqueness is the whole definition, and it is what makes functions useful in engineering: if a circuit is a function of its input, then for each input there is a well-determined output.
| Name | What it is | Example with \( f(x) = \sqrt{x - 1} \) |
|---|---|---|
| Domain | The admitted input values | [1, +∞): there is no real root of a negative number |
| Codomain | The set where the outputs land, by convention R | R |
| Range | The values the function actually reaches | [0, +∞) |
| Graph | The pairs (x, f(x)) drawn in the plane | Half a parabola lying on its side, with its vertex at (1, 0) |
A curve in the plane is the graph of a function if and only if no vertical line crosses it more than once. That is why the circle x² + y² = 1 is not a function: for x = 0 there would be two values, 1 and −1. It is split into two functions, the upper semicircle and the lower one.
02Finding the domain
When a function is given by its formula, the domain is the largest set where that formula makes sense. Three prohibitions cover almost every case:
- Do not divide by zero: the denominator cannot vanish.
- Do not take an even root of a negative number.
- Do not take the logarithm of zero or of a negative number.
For f(x) = ln(x + 2) / √(4 − x²): we need x + 2 > 0, that is x > −2; and 4 − x² > 0 —strict, because it is also in the denominator—, that is −2 < x < 2.
Intersecting: Dom f = (−2, 2).
In an engineering problem the domain is usually narrowed even further for physical reasons: a resistance cannot be negative, nor can a frequency, and time is measured from the moment the switch is closed.
03The families you have to recognize by heart
| Family | Form | Feature | Where it appears in electronics |
|---|---|---|---|
| Polynomial | anxn + … + a₀ | Continuous on all of R | Approximations, calibration-curve fits |
| Rational | Quotient of polynomials | Asymptotes where the denominator vanishes | Transfer functions: zeros and poles |
| Exponential | ax, ex | Grows faster than any power | Charging a capacitor, diode current |
| Logarithmic | logax | Inverse of the exponential | The decibel, Bode scales |
| Trigonometric | sin x, cos x, tan x | Periodic | All of alternating current |
| Piecewise-defined | A different formula in each region | Can have jumps | Amplifier saturation, the step, rectification |
04Properties you can read off the graph
Even if f(−x) = f(x): symmetric about the y-axis, like x² or the cosine. Odd if f(−x) = −f(x): symmetric about the origin, like x³ or the sine. It saves half the work, and in Fourier analysis it decides which harmonics appear.
Increasing if a larger input gives a larger output. A strictly monotonic function is automatically one-to-one, and therefore invertible on its range.
It is bounded if its graph fits between two horizontal lines. A real amplifier is always bounded: this is called saturation.
f(x + T) = f(x) for all x. The smallest positive T is the period, and its reciprocal is the frequency.
A function is one-to-one (injective) if different inputs give different outputs —no horizontal line crosses the graph twice—, onto (surjective) if it reaches the whole codomain, and bijective if it is both. Only bijective functions have an inverse.
05Combining functions
Functions are added, subtracted, multiplied and divided pointwise. And they can also be composed, which is the operation that is specific to them: applying one to the result of the other.
Two circuit stages in cascade are a composition of functions; that is why order matters. And the inverse appears whenever you need to solve for something: from the decibel back to the power ratio, from a sensor reading to the actual temperature.
06Moving and deforming a graph
| Expression | Effect on the graph | In a signal |
|---|---|---|
| f(x) + k | Vertical shift | DC component, offset |
| f(x − a) | Horizontal shift to the right | Delay |
| \( A \cdot f(x) \) | Vertical stretch | Gain |
| \( f(k \cdot x) \) | Horizontal compression if k > 1 | Frequency change |
| \( -f(x) \) and \( f(-x) \) | Reflection about the x-axis · about the y-axis | Phase inversion · time reversal |
A general sinusoid, v(t) = V₀ + A·sin(ωt + φ), is the sine function with all four transformations applied at once. Recognizing them saves you from studying each signal from scratch.
Choose a function and move the four parameters of y = A · f(k(x − a)) + c. The dotted curve is the inverse, mirrored about y = x.
07The static transfer function
The most direct way a function appears in electronics is the transfer characteristic: the output as a function of the input, measured point by point at DC.
- Ideal amplifier: vo = A·vi, a straight line through the origin. It is linear in the strict sense of linear algebra.
- Real amplifier: the same line, but bounded by the supply voltages. It is a piecewise-defined function, and the flat region is saturation. Outside the straight segment superposition no longer holds.
- Diode: i = IS(ev/nVT − 1), a pure exponential. Its being so abrupt is what makes it usable as a switch.
- NTC thermistor: a resistance that decays exponentially with temperature; you have to invert the function to read the temperature from the measured resistance.
- Decibel: a logarithmic function of the power ratio, chosen precisely because it turns products into sums.
08In the lab
With a variable power supply and a multimeter, measure the output of an op amp in a non-inverting configuration for inputs from −3 to 3 V, in steps of 0.25 V. Plot the points, identify the linear segment, measure its slope (the gain) and locate where saturation begins. You can try it out beforehand in the operational amplifiers course.
Write the function that gives the current of a resistive divider as a function of R₂, state its mathematical domain and then its actual physical domain. Explain the difference.
Using the table for an NTC thermistor, fit R(T) and solve for T(R). Check that the composition of the two returns the original value, and discuss over what range the inversion is reliable.
09Common mistakes
- Confusing range with codomain. The range is what is actually reached.
- Forgetting that an even root requires a radicand ≥ 0, and > 0 if it is also in a denominator.
- Writing f−1(x) for 1/f(x). They are different things: one is the inverse, the other the reciprocal.
- Composing in the wrong order: in g∘f, f is applied first.
- Inverting a function that is not one-to-one without first restricting the domain.
- Using an amplifier’s gain outside its linear region.
10Self-assessment
Domain of f(x) = √(x² − 9).
x² − 9 ≥ 0 holds for |x| ≥ 3: (−∞, −3] ∪ [3, +∞).
Is f(x) = x³ − x even, odd or neither?
f(−x) = −x³ + x = −f(x): it is odd.
If f(x) = 2x + 1 and g(x) = x², what are (g∘f)(2) and (f∘g)(2)?
(g∘f)(2) = g(5) = 25 and (f∘g)(2) = f(4) = 9. The order changes the result.
What is the inverse of f(x) = 3x − 6?
y = 3x − 6 → x = (y + 6)/3, so f−1(x) = (x + 6)/3.
Why does a saturated amplifier stop being a linear function?
Because in the flat region the output does not follow the input: doubling the input no longer doubles the output and superposition stops holding.
What graph transformation represents a 2 ms delay in a signal?
A horizontal shift: v(t − 0.002).
11Further reading
- Hebe Rabuffetti. Introducción al análisis matemático (Cálculo 1) (in Spanish). El Ateneo. An Argentine classic, written for the first year of engineering: functions, domains and graphs in great detail.
- James Stewart. Single Variable Calculus: Early Transcendentals. 7th ed. (Spanish edition, “Cálculo de una variable. Trascendentes tempranas”), Cengage Learning, 2012. The most widely used today: an excellent treatment of functions, transformations and models.
- Manuel Sadosky and Rebeca Ch. de Guber. Elementos de cálculo diferencial e integral (in Spanish). Alsina. Another Argentine classic, rigorous and sparing with words.