Pronunciation: [ˈɒd fˈʌŋkʃən] (IPA) 
The term "odd function" in mathematics is spelled as /ɒd ˈfʌŋkʃən/. The word "odd" is pronounced as /ɒd/, with the vowel sound "o" followed by the consonant sound "d". The word "function" is pronounced as /ˈfʌŋkʃən/, with the first syllable pronounced as "fun" and the second syllable pronounced as "k-shun". In mathematics, an "odd function" is a function that satisfies the condition f(-x) = -f(x), where x is any real number.
An odd function refers to a mathematical function that exhibits symmetry across the origin of a coordinate system. It is a type of function that satisfies the property where for any input value x, replacing it with its negative counterpart (-x) produces the opposite signed output value. In other words, if f(x) is an odd function, then f(-x) = -f(x).
Specifically, odd functions are characterized by the property that their graphs are symmetric with respect to the y-axis. This implies that the function's values on one side of the y-axis are mirrored when reflected across the origin. Consequently, an odd function passes through the origin (0, 0) and has a y-intercept of zero.
The distinctive behavior of odd functions provides certain advantages for mathematical analysis. For instance, due to their symmetry, odd functions often possess distinct integration properties. Consequently, when integrating an odd function over a symmetric interval around the origin, the total area under the curve is always zero.
Examples of commonly encountered odd functions include sine (sin(x)), tangent (tan(x)), and reciprocal (1/x) functions. These functions display the characteristic symmetry of odd functions, where their values alternate in sign when substituted with positive or negative inputs. Overall, the concept of an odd function is important in various fields of mathematics, such as calculus, algebra, and differential equations.
The etymology of the word "odd" as used in the context of mathematics, specifically to describe an "odd function", comes from the Old Norse word "othr" which means "second" or "other". This word evolved into the Middle English word "odde", meaning "unparalleled" or "unequal". In mathematics, an odd function refers to a function that exhibits symmetry about the origin and has the property that f(-x) = -f(x) for all x. This concept of "unequal" or "opposite" values on opposite sides of the origin aligns with the use of the term "odd" to describe such functions.
