Domain of a function: Definition & Examples

The domain of a function is the set of inputs for which that function is defined. For a financial model, specifying the domain also means stating which inputs the model admits: a formula may accept a number that makes no sense for the contract being modelled.

If a function is written

f:D\to C

, then

D

is its domain and

C

its codomain, the declared set of possible destinations. The range is the set of outputs actually attained,

f(D)=\{f(x):x\in D\}

, which may be smaller than

C

. Inputs and outputs need not even have the same units: a yield can be the input and a price in dollars the output.

A discounting formula with a real boundary

Suppose a hypothetical contract pays USD 100 exactly half a year from now. Ignore default, fees and intermediate cash flows. Let $y$ be its annual effective yield, entered as a decimal, and explicitly adopt fractional-year compound discounting:

P(y)=\frac{100}{(1+y)^{1/2}}=\frac{100}{\sqrt{1+y}}.

To calculate a real present value, the square root needs

1+y\ge0

. Because that root is also the denominator, zero is excluded. Both restrictions together give

D=\{y\in\mathbb R:y>-1\}=(-1,\infty).

At an annual effective yield of 21%,

P(0.21)=\frac{100}{\sqrt{1.21}}=\frac{100}{1.1} =90.909090\ldots\text{ USD}.

Annual effective yield	Present value (USD)	Domain decision
−36%	125.00	Admitted; negative yield is allowed
0%	100.00	Admitted; no discounting
21%	90.91	Admitted
300%	50.00	Admitted; a hypothetical stress input
−100%	Undefined	Excluded: denominator is zero
Below −100%	No real value under this formula	Excluded: negative square-root argument

Synthetic half-year USD 100 payment: annual effective yields must exceed minus 100 percent. At 21 percent the present value is USD 90.91; the excluded minus 100 percent boundary is shown with a dashed line and an open circle on the input-domain strip.

Original hypothetical example. The plotted curve is a finite window; the domain extends to every finite yield above −100%.

Here the range is

(0,\infty)

in USD: every positive price can be reached by choosing

y=(100/P)^2-1

. Zero is never an output at a finite admitted yield. As

y

approaches −1 from above, the price grows without bound; that one-sided limit does not give a value at −1.

The boundary belongs to this convention. With continuous compounding,

P(r)=100e^{-r/2}

admits every finite real continuously compounded rate

r

. Do not transfer the −100% boundary to that different rate variable. Nor does a mathematically admitted yield establish that a quoted market price or investment is sensible.

Find the domain before simplifying

Unless a domain is stated, an elementary real-valued formula usually takes its natural domain: every real input for which the expression is defined. The main restrictions are:

Operation	Required condition
Division	Denominator is nonzero
Even root, including a square root	Radicand is nonnegative
Real logarithm	Argument is strictly positive

These conditions apply simultaneously. For example,

h(x)=\frac{\sqrt{x-2}}{x-5} \quad\Longrightarrow\quad D_h=[2,5)\cup(5,\infty).

The input 2 is included: a zero numerator is harmless here, and $h(2)=0$ . The input 5 is excluded. Square brackets include a finite endpoint; parentheses exclude it. Infinity is not a real endpoint and always takes a parenthesis.

Algebraic cancellation preserves values only where the original expression existed:

\frac{x^2-9}{x-3}=x+3\qquad\text{for }x\ne3.

The original rational function still excludes 3. Defining its value there to be 6 creates an extension, rather than recovering a value the fraction already had.

For a composite function, check both stages:

D_{f\circ g}=\{x\in D_g:g(x)\in D_f\}.

For instance, $\ln(1-x^2)$ requires $1-x^2>0$ , so its domain is $(-1,1)$ . The inner polynomial accepts all real inputs, but the outer logarithm rejects zero and negative intermediate values.

A model can impose a smaller domain

Suppose $V(q)=50q$ describes the USD value of a long-only holding priced at USD 50 per share, with whole shares only. The formula accepts every real $q$ . This holding model admits only $q\in\{0,1,2,\ldots\}$ . Twenty shares give USD 1,000; a negative share count would describe a short position outside the stated model, and a fractional count would violate its whole-share assumption.

Likewise, $100/(1+y)^2$ is algebraically defined for $y<-1$ , because the exponent is an integer. A standard positive-growth-factor discounting model can still explicitly require $1+y>0$ . The formula's natural domain and the model's chosen domain answer different questions.

In a pricing or risk calculation, state the input's meaning and units, find the formula's restrictions, then intersect them with the contract and model restrictions. A failed domain check means the stated function has no value there; a nearby limit, a simplified expression or a software-produced number cannot silently change that contract.

Sources

James Stewart, Single Variable Calculus: Early Transcendentals, 6th edition (2008), §§1.1 and 1.3: domains, ranges, natural domains and combinations of functions.
John C. Hull, Options, Futures, and Other Derivatives, 9th Global Edition, §4.2: annual effective and continuously compounded interest-rate conventions.
OpenStax, Calculus Volume 1, §1.1: Review of Functions: function notation and admissible real inputs.

A discounting formula with a real boundary

Find the domain before simplifying

A model can impose a smaller domain

Further reading

Sources

Related terms

Composite function

Logarithm

Mathematical function

One-sided limit

Present value

Range of a function

Rational function