Inverse Functions
Bayani Gaba-gaba
You already know how a function turns an input into an output. An inverse function does the reverse: it takes the output back to the input it came from. Think of it as the undo button for a function. If one machine doubles a number and adds three, the inverse machine must subtract three and then halve, exactly reversing each step in the opposite order.
But not every function can be undone. In this lesson you will learn the crucial test of whether an inverse even exists, why a function must be one-one to have one, and the reliable swap-and-rearrange method for finding the inverse. These ideas sit at the heart of the Functions section and reward careful, tidy algebra.
Manufura
- Explain in words why a given function does not have an inverse.
- Find the inverse of a one-one function and use correct notation.
Takardar Darasi
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Barka da kammala darasi akan Inverse Functions. Yanzu da kuka bincika mahimman raayoyi da raayoyi, lokaci yayi da zaku gwada ilimin ku. Wannan sashe yana ba da ayyuka iri-iri Tambayoyin da aka tsara don ƙarfafa fahimtar ku da kuma taimaka muku auna fahimtar ku game da kayan.
Za ka gamu da haɗe-haɗen nau'ikan tambayoyi, ciki har da tambayoyin zaɓi da yawa, tambayoyin gajeren amsa, da tambayoyin rubutu. Kowace tambaya an ƙirƙira ta da kyau don auna fannoni daban-daban na iliminka da ƙwarewar tunani mai zurfi.
Yi wannan ɓangaren na kimantawa a matsayin wata dama don ƙarfafa fahimtarka kan batun kuma don gano duk wani yanki da kake buƙatar ƙarin karatu. Kada ka yanke ƙauna da duk wani ƙalubale da ka fuskanta; maimakon haka, ka kallesu a matsayin damar haɓaka da ingantawa.
- What is the inverse of f(x) = 2x + 3? A. (x - 3)/2 B. (x + 3)/2 C. 2x - 3 D. 1/(2x + 3) Answer: A
- Which type of function always has an inverse? A. Many-one B. One-one C. Quadratic over all reals D. Constant Answer: B
- Why does g(x) = x^2 (for all real x) have no inverse? A. It is undefined at 0 B. It is one-one C. It is many-one, since 2 and -2 both give 4 D. It has no range Answer: C
- The inverse of f(x) = x/4 - 5 is: A. 4x - 5 B. 4(x + 5) C. (x - 5)/4 D. 1/(x/4 - 5) Answer: B
- If f(4) = 11, what is f^{-1}(11)? A. 11 B. 1/11 C. 4 D. 7 Answer: C
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