--- title: Inverse Functions Explained - Definition & Example description: "Inverse function, explained: when a function can be undone, and how to write the inverse." url: https://astra-ai.co/videos/functions/inverse-function/inverse-function-explanation/ locale: en type: video-lesson keywords: - inverse function - undo a function - one-to-one - swap coordinates published: 2026-10-01T10:08:15.908Z updated: 2026-10-04T22:00:13.011Z alternates: bg: https://astra-ai.co/bg/videos/funktsii/obratna-funktsiya/obratna-funktsiya-obyasnenie/ br: https://astra-ai.co/br/videos/funcoes/capitulo-matematica-044/video-matematica-004/ cs: https://astra-ai.co/cs/videos/funkce/inverzni-funkce/inverzni-funkce-vysvetleni/ da: https://astra-ai.co/da/videos/funktioner/invers-funktion/invers-funktion-forklaring/ de: https://astra-ai.co/de/videos/funktionen/inverse-funktion-kapitel/inverse-funktion-erklrung/ el: https://astra-ai.co/el/videos/synartiseis/antistrofi-synartisi/inverse-function-explanation/ en: https://astra-ai.co/videos/functions/inverse-function/inverse-function-explanation/ en-IN: https://astra-ai.co/en-IN/videos/functions/chapter-19891853/inverse-function-explanation/ en-PH: https://astra-ai.co/en-PH/videos/functions/chapter-19891853/lesson-d844bc09/ es: https://astra-ai.co/es/videos/funciones/funcion-inversa-que-es-como-encontrarla-y-ejemplos/funcion-inversa-explicacion-propiedades-y-ejemplo/ es-AR: https://astra-ai.co/es-AR/videos/funciones/capitulo-38/leccion-4/ es-CL: https://astra-ai.co/es-CL/videos/funciones/capitulo-38/funcion-inversa-explicacion-propiedades-y-ejemplo/ es-PE: https://astra-ai.co/es-PE/videos/funciones/capitulo-video-19891853/leccion-de-video-d844bc09/ et: https://astra-ai.co/et/videos/funktsioonid/poordfunktsioon/poordfunktsioon-selgitus/ fi: https://astra-ai.co/fi/videos/funktiot/kaanteisfunktio/kaanteisfunktio-selitys/ fr: https://astra-ai.co/fr/videos/fonction-mathematique/fonction-reciproque/fonction-reciproque-explication/ hr: https://astra-ai.co/hr/videos/funkcije/inverzna-funkcija/inverzna-funkcija-objasnjenje/ hu: https://astra-ai.co/hu/videos/fuggvenyek/inverz-fuggveny/inverz-fuggveny-magyarazat/ id: https://astra-ai.co/id/videos/fungsi/fungsi-invers/fungsi-invers-penjelasan/ it: https://astra-ai.co/it/videos/funzioni/funzione-inversa/funzione-inversa-spiegazione/ ja: https://astra-ai.co/ja/videos/functions-ja/38/004/ km: https://astra-ai.co/km/videos/functions/chapter-19891853/lesson-d844bc09/ ko: https://astra-ai.co/ko/videos/functions-ko/38/004/ lo: https://astra-ai.co/lo/videos/lo-a529886c/lo-19891853/d844bc09/ lt: https://astra-ai.co/lt/videos/funkcijos/atvirkstine-funkcija/pamoka-d844bc09/ lv: https://astra-ai.co/lv/videos/funkcijas/apgriezta-funkcija/stunda-d844bc09/ ms: https://astra-ai.co/ms/videos/fungsi/bab-44-matematik-asas/pelajaran-004/ ms-BN: https://astra-ai.co/ms-BN/videos/fungsi/bab-19891853/pelajaran-d844bc09/ mx: https://astra-ai.co/mx/videos/funciones/capitulo-38/leccion-4/ my: https://astra-ai.co/my/videos/my-a529886c/my-19891853/d844bc09/ nl: https://astra-ai.co/nl/videos/functies/inverse-functie/inverse-functie-uitleg/ no: https://astra-ai.co/no/videos/funksjoner/invers-funksjon/invers-funksjon-forklaring/ pl: https://astra-ai.co/pl/videos/funkcje/funkcja-odwrotna/funkcja-odwrotna-wyjasnienie/ pt: https://astra-ai.co/pt/videos/funcoes/funcao-inversa/funcao-inversa-explicacao/ pt-TL: https://astra-ai.co/pt-TL/videos/funcoes/capitulo-de-video-038/licao-de-video-004/ ro: https://astra-ai.co/ro/videos/functii/functie-inversa/lectie-video-d844bc09/ sk: https://astra-ai.co/sk/videos/funkcie/kapitola-038/lekcia-004/ sl: https://astra-ai.co/sl/videos/funkcije/inverzna-funkcija/inverzna-funkcija-razlaga/ sr: https://astra-ai.co/sr/videos/funkcije/poglavlje-038/lekcija-004/ sv: https://astra-ai.co/sv/videos/funktioner/invers-funktion/invers-funktion-forklaring/ th: https://astra-ai.co/th/videos/fangchan/bot-44-fangchan-phakphan/botrian-004/ tr: https://astra-ai.co/tr/videos/fonksiyonlar/bolum-038/ders-004/ uk: https://astra-ai.co/uk/videos/uk-topic-19/uk-chapter-44/uk-inverse-function-explanation/ vi: https://astra-ai.co/vi/videos/vi-topic-19/vi-chapter-44/vi-inverse-function-explanation/ zh-HK: https://astra-ai.co/zh-HK/videos/shuxue-kecheng-019/shuxue-zhang-044/shuxue-kecheng-004/ zh-TW: https://astra-ai.co/zh-TW/videos/functions/chapter-19891853/lesson-d844bc09/ --- # Inverse Function - Explanation > An inverse function undoes the original function. This lesson explains when that inverse exists and how to write it. An inverse function is a key concept in mathematics that allows for understanding the relationships between functions and their "reversals." This concept is particularly important as it broadens our understanding of functions and their applications. ## DEFINITION It is important to mention that a function has an inverse function only if it is **bijective**. This means it must be both: - **INJECTIVE** (one-to-one): Each element in the domain maps to a unique element in the codomain (different inputs give different outputs). - **SURJECTIVE** (onto): Every element in the codomain is mapped to by at least one element from the domain (all possible outputs are reached). If a function f maps x to y (f(x) = y), its inverse function, denoted f^-1(x), maps y back to x (f^-1(y) = x). ## PROPERTIES - **GRAPH SYMMETRY:** The graph of an inverse function f^-1(x) is symmetrical to the graph of the function f(x) across the line y = x. - **DOMAIN AND RANGE:** The domain of the inverse function f^-1(x) is the range of the original function f(x), and the range of f^-1(x) is the domain of f(x). - **INVERSE OF THE INVERSE:** If you have a function f(x) and its inverse function f^-1(x), the inverse of f^-1(x) is the original function f(x). That is, (f^-1)^-1(x) = f(x). ## EXAMPLE For a better understanding, let's look at an example. Let the given function be f(x) = 2x + 3. This function is bijective, which means it has an inverse function. To find it: - Replace f(x) with y: y = 2x + 3. - Swap x and y to represent the inverse relationship: x = 2y + 3. - Solve for y: x - 3 = 2y y = (x - 3) / 2 Therefore, the inverse function is f^-1(x) = (x - 3) / 2. ## CONCLUSION Invertibility is essential for understanding how functions can be "undone" or "reversed." Understanding these functions opens the door to a deeper comprehension of mathematical structures and allows for a better grasp of various mathematical concepts, such as solving equations and transforming graphs. Inverse functions are fundamental in algebra, analysis, and numerous other areas of mathematics.