"1.2.22 Inverse and composite functions (A-level only) Flashcards","topicId":"0c2ac726-6c6e-47de-9ad4-4899bc05a022","topicSlug":"inverse-and-composite-functions","cards":[{"id":"2cd9a166-2d99-421f-9503-bf03606c8a5d","cardType":"basic","front":"In $f(g(x))$, which function acts first?","back":"Apply $g$ first.","points":1,"mcqOptions":null},{"id":"2c0f4da9-7530-4eae-9689-4270bb3ee05f","cardType":"basic","front":"What condition must hold for $f(g(x))$ to exist?","back":"$x$ is in the domain of $g$ and $g(x)$ is in the domain of $f$.","points":2,"mcqOptions":null},{"id":"870098a2-ecdc-4454-89f8-9789fb844c6a","cardType":"basic","front":"Why is $f(g(x))$ generally different from $g(f(x))$?","back":"The functions are applied in a different order.","points":2,"mcqOptions":null},{"id":"e4bc7189-d65c-4f0c-8911-97651871af34","cardType":"basic","front":"If $f(x)=2x+3$ and $g(x)=x^2-1$, find $f(g(x))$.","back":"$f(g(x))=2x^2+1$","points":2,"mcqOptions":null},{"id":"7e3a4470-fc22-4aa1-a4a5-d84b123d0132","cardType":"basic","front":"If $f(x)=2x+3$ and $g(x)=x^2-1$, find $g(f(x))$.","back":"$g(f(x))=4x^2+12x+8$","points":2,"mcqOptions":null},{"id":"47757027-ef3c-49cd-ae07-4870299ecb6c","cardType":"basic","front":"Find the domain of $f(g(x))$ when $f(x)=\\sqrt{x}$ and $g(x)=3-x$.","back":"$x\\leq 3$","points":2,"mcqOptions":null},{"id":"2eebda7f-b3ee-47ec-b7b1-b7a7532ca8a7","cardType":"cloze","front":"An inverse function [ ] the original function, whereas $\\frac{1}{f(x)}$ is its reciprocal.","back":"An inverse function reverses the original function, whereas $\\frac{1}{f(x)}$ is its reciprocal.","points":null,"mcqOptions":null},{"id":"7b3c997c-53e4-41f1-a3f9-fd3e3f932da2","cardType":"cloze","front":"An inverse function reverses the original function, whereas $\\frac{1}{f(x)}$ is its [ ].","back":"An inverse function reverses the original function, whereas $\\frac{1}{f(x)}$ is its reciprocal.","points":null,"mcqOptions":null},{"id":"5b6afdad-ef4c-4278-bc2b-f834c88a871c","cardType":"basic","front":"What property must a function have for its inverse to be a function?","back":"It must be one-to-one.","points":1,"mcqOptions":null},{"id":"d02290cc-e007-4e54-832d-8cd4b980ed57","cardType":"basic","front":"How does the horizontal line test identify a one-to-one function?","back":"Every horizontal line meets the graph at most once.","points":2,"mcqOptions":null},{"id":"9472d632-2f6b-4116-916c-cb1fa331dc6a","cardType":"basic","front":"Why is $f(x)=x^2$ not one-to-one on all real numbers?","back":"Different inputs can give the same output, such as $f(2)=f(-2)$.","points":2,"mcqOptions":null},{"id":"32c62965-7f5d-40e0-8531-c08464440b5a","cardType":"basic","front":"How can a many-to-one function often be made invertible?","back":"Restrict its domain to an entirely increasing or decreasing interval.","points":2,"mcqOptions":null},{"id":"59ecf9db-67f9-4f4c-b279-48784dc21943","cardType":"basic","front":"For $f(x)=x^2$ restricted to $x\\geq0$, what is $f^{-1}(x)$?","back":"$f^{-1}(x)=\\sqrt{x}$","points":1,"mcqOptions":null},{"id":"a0176533-6018-4563-8894-09aefe313bdd","cardType":"cloze","front":"The domain of $f^{-1}$ is the [ ], while the range of $f^{-1}$ is the domain of $f$.","back":"The domain of $f^{-1}$ is the range of $f$, while the range of $f^{-1}$ is the domain of $f$.","points":null,"mcqOptions":null},{"id":"bc715433-ec86-4a53-a16b-cd05735866f7","cardType":"cloze","front":"The domain of $f^{-1}$ is the range of $f$, while the range of $f^{-1}$ is the [ &n