section_path: Specifying a function/By a formula
section_name: By a formula
paragraph_text: By a formula
  
  Functions are often defined by an expression that describes a combination of arithmetic operations and previously defined functions; such a formula allows computing the value of the function from the value of any element of the domain.For example, in the above example, f {\displaystyle f} can be defined by the formula f (n) = n + 1 {\displaystyle f(n)=n+1}, for n ∈ { 1, 2, 3 } {\displaystyle n\in \{1,2,3\}}.
  
  When a function is defined this way, the determination of its domain is sometimes difficult. If the formula that defines the function contains divisions, the values of the variable for which a denominator is zero must be excluded from the domain; thus, for a complicated function, the determination of the domain passes through the computation of the zeros of auxiliary functions. Similarly, if square roots occur in the definition of a function from R {\displaystyle \mathbb {R} } to R, {\displaystyle \mathbb {R},} the domain is included in the set of the values of the variable for which the arguments of the square roots are nonnegative.
  
  For example, f (x) = 1 + x 2 {\displaystyle f(x)={\sqrt {1+x^{2}}}} defines a function f: R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } whose domain is R, {\displaystyle \mathbb {R},} because 1 + x 2 {\displaystyle 1+x^{2}} is always positive if x is a real number. On the other hand, f (x) = 1 − x 2 {\displaystyle f(x)={\sqrt {1-x^{2}}}} defines a function from the reals to the reals whose domain is reduced to the interval. (In old texts, such a domain was called the domain of definition of the function.)
  
  Functions can be classified by the nature of formulas that define them: