Difference between revisions of "Domain"
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| [[pointwise product of functions|pointwise product]] || <math>f \cdot g</math> (sometimes denoted <math>fg</math>) is the function <math>x \mapsto f(x)g(x)</math><br><math>f_1 \cdot f_2 \cdot \dots f_n</math> (sometimes denoted <math>f_1f_2\dots f_n</math> is the function <math>x \mapsto f_1(x)f_2(x) \dots f_n(x)</math> || Intersection of the domains of all the functions being multiplied || Similar to the note for sums.<br> <toggledisplay>Note that this definition is quasi-paradoxical, in that the function defined as <math>f \cdot g</math> where <math>f(x) = x - 2</math> and <math>g(x) = 1/(x - 2)</math> is 1 everywhere ''except'' at 2. This is because even though the expressions for <math>f</math> and <math>g</math> cancel each other, evaluating at <math>x = 2</math> does not make sense for the expression ''prior'' to algebraic simplication by cancellation.</toggledisplay> | | [[pointwise product of functions|pointwise product]] || <math>f \cdot g</math> (sometimes denoted <math>fg</math>) is the function <math>x \mapsto f(x)g(x)</math><br><math>f_1 \cdot f_2 \cdot \dots f_n</math> (sometimes denoted <math>f_1f_2\dots f_n</math> is the function <math>x \mapsto f_1(x)f_2(x) \dots f_n(x)</math> || Intersection of the domains of all the functions being multiplied || Similar to the note for sums.<br> <toggledisplay>Note that this definition is quasi-paradoxical, in that the function defined as <math>f \cdot g</math> where <math>f(x) = x - 2</math> and <math>g(x) = 1/(x - 2)</math> is 1 everywhere ''except'' at 2. This is because even though the expressions for <math>f</math> and <math>g</math> cancel each other, evaluating at <math>x = 2</math> does not make sense for the expression ''prior'' to algebraic simplication by cancellation.</toggledisplay> | ||
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− | | [[pointwise quotient of functions|pointwise quotient]] || <math>f/g</math> is the function <math>x \mapsto f(x)/g(x)</math> || Intersection of the domain of <math>f</math> with the ''subset'' of the domain of <math>g</math> comprising those points <math>x</math> where <math>g(x) \ne 0</math>. | + | | [[pointwise quotient of functions|pointwise quotient]] || <math>f/g</math> is the function <math>x \mapsto f(x)/g(x)</math> || Intersection of the domain of <math>f</math> with the ''subset'' of the domain of <math>g</math> comprising those points <math>x</math> where <math>g(x) \ne 0</math>. || |
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| [[composite of two functions]] || <math>f \circ g</math> is the function <math>x \mapsto f(g(x))</math> || Set of those values <math>x</math> for which <math>g(x)</math> lies inside the domain of <math>f</math>. || | | [[composite of two functions]] || <math>f \circ g</math> is the function <math>x \mapsto f(g(x))</math> || Set of those values <math>x</math> for which <math>g(x)</math> lies inside the domain of <math>f</math>. || |
Latest revision as of 00:43, 29 April 2014
This page lists a core term of calculus. The term is used widely, and a thorough understanding of its definition is critical.
See a complete list of core terminology
Contents
Definition
General definition
The domain of a function is the set of inputs allowed for the function, i.e., the set of values that can be fed into the function to give a valid output.
If is a function, the domain of is the set .
For a function described by an expression or procedure without explicit domain specification
If a function of one variable (i.e., a function whose domain is a subset of the reals) is defined by means of an expression or procedure, but the domain is not explicitly specified, the domain is taken to be the maximum possible domain, i.e., the largest subset of the reals on which that expression or procedure makes sense and yields a meaningful answer.
On the other hand, the domain of the function may be specified explicitly to be a proper subset of the maximum possible domain that we could infer from the expression alone.
Caveats
Domain and restriction of domain
The study of a function depends crucially on the domain on which the function is being studied. If a function of one variable is defined solely by means of an expression or procedure, the domain of the function is taken to be the largest possible subset of the reals on which that expression or procedure makes sense and gives a valid answer. However, we can also consider functions restricted to domains that are strictly smaller than the maximum possible domain on which the expression being used for the function makes sense. The behavior of the function, as well as answers to questions like whether it is increasing or decreasing and what its extreme values are, depends on what domain we are considering the function on.
Here are some examples:
- Consider the function defined as follows: is defined as the area of a circle with diameter . Ignoring the boundary case of point circles and line circles, the only possible inputs for this function are positive reals, so is a function from the positive reals to the positive reals given by the expression . However, if we look only at the expression for , then that expression makes sense for all real numbers, including zero and negative real numbers as well as positive real numbers. Call the latter function , i.e., for all . Then, is the restriction of to the subdomain . Note that:
- is not an increasing function, but the restriction is an increasing function.
- is not a one-one function, but the restriction is a one-one function.
- attains its absolute minimum value, but the restriction does not.
Computation of domain
Computation of maximum possible domain from algebraic expression
We discuss below how we can compute the domain of a function obtained by using a pointwise combination, composite, inverse function, and piecewise definition:
Method for constructing new functions from old | In symbols | Domain of the new function in terms of domains of the old functions | Additional notes |
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pointwise sum | is the function is the function |
Intersection of the domains of all the functions being added | [SHOW MORE] |
pointwise difference | is the function | Intersection of the domains of the functions being subtracted | Similar to the note for sums |
scalar multiple by a constant | is the function where is a real number | Same as the domain of the original function | |
pointwise product | (sometimes denoted ) is the function (sometimes denoted is the function |
Intersection of the domains of all the functions being multiplied | Similar to the note for sums. [SHOW MORE] |
pointwise quotient | is the function | Intersection of the domain of with the subset of the domain of comprising those points where . | |
composite of two functions | is the function | Set of those values for which lies inside the domain of . | |
inverse function of a one-one function | sends to the unique such that | Same as the range of the original one-one function | |
piecewise definition | Fill this in later | Union of the domain of definition for each piece. This domain is usually given explicitly or requires finding the subset of an explicitly specified set where an explicit expression makes sense. |