https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:composite/x9e81a4f98389efdf:invertible/a/intro-to-invertible-functions 2026-06-16T15:55:43.561813682Z Intro to invertible functions Not all functions have inverses. Those that do are called "invertible." Learn how we can tell whether a function is invertible or not. article https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:composite/x9e81a4f98389efdf:inverse-functions-in-graphs-and-tables/v/reading-inverse-values-from-a-graph 2023-07-18T22:48:45.033562364Z Reading inverse values from a graph Learn how to find the inverse of a function and how to graph it in this video. The inverse function swaps the x and y values of the original function, and the graph of this inverse function is a reflection of the original function about the line y=x. Sal Khan video https://cdn.kastatic.org/ka_thumbnails_cache/2f7debb4-e9ae-441b-8631-08dacf15f361_1280_720_base.png Reading inverse values from a graph Learn how to find the inverse of a function and how to graph it in this video. The inverse function swaps the x and y values of the original function, and the graph of this inverse function is a reflection of the original function about the line y=x. https://cdn.kastatic.org/ka-youtube-converted/nEWYKpTCX-U.mp4/nEWYKpTCX-U.mp4 254 Inverse functions https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:composite/x9e81a4f98389efdf:inverse-functions-in-graphs-and-tables/v/reading-inverse-values-from-a-table 2022-10-07T00:22:18.752150461Z Reading inverse values from a table An inverse function takes the output from the original function and gives us the input. Given a table of inputs and outputs for a function, evaluate the inverse function for a certain input. Sal Khan video https://cdn.kastatic.org/ka_thumbnails_cache/559e3de8-149a-4fba-8257-3e6d62fa0067_1280_720_base.png Reading inverse values from a table An inverse function takes the output from the original function and gives us the input. Given a table of inputs and outputs for a function, evaluate the inverse function for a certain input. https://cdn.kastatic.org/ka-youtube-converted/ygVpyyASavk.mp4/ygVpyyASavk.mp4 128 Inverse functions https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:composite/x9e81a4f98389efdf:inverse-functions-in-graphs-and-tables/e/inverse-functions-graphs-and-tables 2026-03-26T05:20:35.16058035Z Inverse functions: graphs and tables Read values of an inverse function from a graph or a table of the original function. exercise https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:composite/x9e81a4f98389efdf:invertible/v/determining-if-a-function-is-invertible 2023-07-18T22:48:45.033562364Z Determining if a function is invertible Learn how to build a mapping diagram for a finite function, and how to use this diagram to determine if the function is invertible. An invertible function has a one-to-one mapping between its domain and range. Functions that map multiple domain elements to the same range element are not invertible. video https://cdn.kastatic.org/ka_thumbnails_cache/f2547404-81e7-482d-b049-13297674c2ac_1280_720_base.png Determining if a function is invertible Learn how to build a mapping diagram for a finite function, and how to use this diagram to determine if the function is invertible. An invertible function has a one-to-one mapping between its domain and range. Functions that map multiple domain elements to the same range element are not invertible. https://cdn.kastatic.org/ka-youtube-converted/dqCTAHHza10.mp4/dqCTAHHza10.mp4 244 Inverse functions https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:composite/x9e81a4f98389efdf:invertible/e/inverse-domain-range 2026-05-05T18:10:50.367167117Z Determine if a function is invertible Given the table of values of a function, determine whether it is invertible or not. Stacy Beaudoin exercise https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:trig/x9e81a4f98389efdf:inverse-trig/v/restricting-trig-function-domain 2021-05-05T14:53:20Z Restricting domains of functions to make them invertible Sal is given the graph of a trigonometric function, and he discusses ways in which he can change the function to make it invertible. Sal Khan video https://cdn.kastatic.org/googleusercontent/oCkVpnSYRrSyQLv70yydN1Eqf4hSaC0ePtyS9lq0aBSMQRVhlyOd5ubTOpIPBb_o7KFTiqGLHUpPrkGMMu5H_T1i Restricting domains of functions to make them invertible Sal is given the graph of a trigonometric function, and he discusses ways in which he can change the function to make it invertible. https://cdn.kastatic.org/ka-youtube-converted/tCV9VyIIaw0.mp4/tCV9VyIIaw0.mp4 360 Inverse functions https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:composite/x9e81a4f98389efdf:invertible/e/restrict-the-domains-of-functions 2026-05-05T18:10:50.367167117Z Restrict domains of functions to make them invertible Given the graph of a function, find ways to restrict its domain in order to make it invertible. Tomer Gal exercise https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:composite/x9e81a4f98389efdf:verifying-inverse/a/verifying-inverse-functions-by-composition 2021-06-30T17:32:36Z Verifying inverse functions by composition Learn how to verify whether two functions are inverses by composing them. For example, are f(x)=5x-7 and g(x)=x/5+7 inverse functions? article https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:composite/x9e81a4f98389efdf:verifying-inverse/v/verify-inverse-functions-from-tables 2022-10-07T00:22:18.752150461Z Verifying inverse functions from tables Inverse functions undo each other. Functions s and t are inverses if and only if s(t(x))=x and t(s(x))=x for every x-value in the domains. Sal Khan video https://cdn.kastatic.org/ka_thumbnails_cache/a0748b15-0c0b-4d80-83e0-7d7a8ee48dfe_1280_720_base.png Verifying inverse functions from tables Inverse functions undo each other. Functions s and t are inverses if and only if s(t(x))=x and t(s(x))=x for every x-value in the domains. https://cdn.kastatic.org/ka-youtube-converted/UTUuyCXFAs4.mp4/UTUuyCXFAs4.mp4 272 Inverse functions https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:composite/x9e81a4f98389efdf:verifying-inverse/v/verifying-function-inverses-by-composition 2022-10-07T00:22:18.752150461Z Verifying inverse functions by composition Sal composes f(x)=(x+7)³-1 and g(x)=∛(x+1)-7, and finds that f(g(x))=g(f(x))=x, which means the functions are inverses! video https://cdn.kastatic.org/ka_thumbnails_cache/28d20992-ce4a-441e-8355-9eac46f8d813_1280_720_base.png Verifying inverse functions by composition Sal composes f(x)=(x+7)³-1 and g(x)=∛(x+1)-7, and finds that f(g(x))=g(f(x))=x, which means the functions are inverses! https://cdn.kastatic.org/ka-youtube-converted/8GEGnSEJA2s.mp4/8GEGnSEJA2s.mp4 401 Inverse functions https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:composite/x9e81a4f98389efdf:verifying-inverse/v/composing-non-inverse-functions 2022-10-07T00:22:18.752150461Z Verifying inverse functions by composition: not inverse Sal composes f(x)=2x-3 and g(x)=½x+3, and finds that f(g(x)) ≠ g(f(x)) ≠ x, which means the functions are not inverses. video https://cdn.kastatic.org/ka_thumbnails_cache/43938826-7235-46f2-94a0-f25070b66ed1_1280_720_base.png Verifying inverse functions by composition: not inverse Sal composes f(x)=2x-3 and g(x)=½x+3, and finds that f(g(x)) ≠ g(f(x)) ≠ x, which means the functions are not inverses. https://cdn.kastatic.org/ka-youtube-converted/PJIL0kRHND0.mp4/PJIL0kRHND0.mp4 252 Inverse functions https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:composite/x9e81a4f98389efdf:verifying-inverse/e/inverses_of_functions 2025-06-25T21:21:38.768284711Z Verify inverse functions Determine whether two functions are inverses or not by composing the functions algebraically, composing them using tables, or evaluating inputs to the composition. Charlotte Auen exercise https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:eq/x2ec2f6f830c9fb89:rational-eq/v/rational-function-inverse 2022-10-06T23:23:50.639907812Z Finding inverses of rational functions The inverse of a function ƒ is a function that maps every output in ƒ's range to its corresponding input in ƒ's domain. We can find an expression for the inverse of ƒ by solving the equation 𝘹=ƒ(𝘺) for the variable 𝘺. See how it's done with a rational function. video https://cdn.kastatic.org/ka_thumbnails_cache/e3404c5d-adea-4350-9099-bee1b7844b6a_1280_720_base.png Finding inverses of rational functions The inverse of a function ƒ is a function that maps every output in ƒ's range to its corresponding input in ƒ's domain. We can find an expression for the inverse of ƒ by solving the equation 𝘹=ƒ(𝘺) for the variable 𝘺. See how it's done with a rational function. https://cdn.kastatic.org/ka-youtube-converted/YI30Ecfl4tc.mp4/YI30Ecfl4tc.mp4 257 Inverse functions https://www.khanacademy.org/math/math-1-illustrative-math-aligned/x193529144fa1e1f6:functions/x193529144fa1e1f6:finding-and-interpreting-inverse-functions/v/finding-and-interpreting-inverse-functions 2026-01-28T16:50:37.680803152Z Finding and interpreting inverse functions Find and interpret an inverse function in terms of situations. Sal Khan video https://cdn.kastatic.org/ka_thumbnails_cache/52b89296-bcf3-4d48-829f-97be2c6cdab2_1280_720_base.png Finding and interpreting inverse functions Find and interpret an inverse function in terms of situations. https://cdn.kastatic.org/ka-youtube-converted/fvJZWSrdSOM.mp4/fvJZWSrdSOM.mp4 178 Inverse functions https://www.khanacademy.org/math/ap-precalculus/xe322600f784e4778:exponential-and-logarithmic-functions/xe322600f784e4778:find-the-inverse-function/e/find-the-inverse 2026-06-16T19:11:42.716495944Z Find the inverse (algebraically) Students will find the inverse function of linear, polynomial, radical and rational functions algebraically. Courtney Chernin-Hitchings exercise