nd functions (A-level only) Flashcards","topicId":"869b3575-660f-4e6f-9bac-e03048cea025","topicSlug":"algebra-and-functions-21","cards":[{"id":"67d9b5fd-8b03-4d6a-85e0-0d05a1c4d68d","cardType":"cloze","front":"The set of permitted input values is the [ ].","back":"The set of permitted input values is the domain.","points":null,"mcqOptions":null},{"id":"a099aabb-44c7-4feb-9aa2-39d9cdd2cbef","cardType":"cloze","front":"The set of possible output values is the [ ].","back":"The set of possible output values is the range.","points":null,"mcqOptions":null},{"id":"a02343e1-c73a-49bb-b370-00f4240d06ec","cardType":"basic","front":"What is the range of $f(x)=x^2+1$ when $x\\geq 0$?","back":"The range is $f(x)\\geq 1$.","points":1,"mcqOptions":null},{"id":"b0d3964a-ac33-458e-b035-439c95b34752","cardType":"basic","front":"A function multiplies by 3 and then adds 5. What does its inverse do?","back":"It subtracts 5, then divides by 3.","points":2,"mcqOptions":null},{"id":"96930293-fbf0-48cf-82d1-f6d909d418b5","cardType":"basic","front":"What is the difference between $f^{-1}(x)$ and $\\frac{1}{f(x)}$?","back":"$$f^{-1}(x)$ is the inverse function; $\\frac{1}{f(x)}$ is the reciprocal.","points":2,"mcqOptions":null},{"id":"19e96f50-0379-4421-84f2-a3585f08fefa","cardType":"cloze","front":"Inverse functions satisfy $\\text{[...]}$.","back":"Inverse functions satisfy $\\boxed{f^{-1}(f(x))=x}$.","points":null,"mcqOptions":null},{"id":"73474043-e3e6-4c45-8989-f199148421fb","cardType":"cloze","front":"A function is one-to-one if $f(a)=f(b)$ implies $\\text{[...]}$.","back":"A function is one-to-one if $f(a)=f(b)$ implies $\\boxed{a=b}$.","points":null,"mcqOptions":null},{"id":"e55c3ca9-d8f5-447d-9930-b743dba57cd9","cardType":"basic","front":"What result in the horizontal line test shows that a function is one-to-one?","back":"Every horizontal line meets the graph at most once.","points":1,"mcqOptions":null},{"id":"0c426375-c9ec-4131-8d9a-26a1c8a12f4a","cardType":"basic","front":"Why is $f(x)=x^2$ not invertible over all real numbers?","back":"Different inputs can give the same output, such as $f(2)=f(-2)$.","points":2,"mcqOptions":null},{"id":"883c1e55-c625-4f76-a50c-6d2882e67fc3","cardType":"basic","front":"What inverse results when $f(x)=x^2$ is restricted to $x\\geq 0$?","back":"$f^{-1}(x)=\\sqrt{x}$.","points":1,"mcqOptions":null},{"id":"79ec6ede-8d19-4f47-8ca5-81d1760fe0c9","cardType":"basic","front":"Why does restricting $f(x)=(x-2)^2+3$ to $x\\geq 2$ make it invertible?","back":"The restriction keeps only the increasing branch, so each output has one input.","points":2,"mcqOptions":null},{"id":"ab4b7182-bedc-4316-b0fc-b96a484fb7bb","cardType":"basic","front":"After writing $y=f(x)$, what key rearrangement is used to find the inverse?","back":"Make $x$ the subject, then swap $x$ and $y$.","points":2,"mcqOptions":null},{"id":"10506151-2ee7-44a5-865a-2937d5af8fcc","cardType":"basic","front":"Find the inverse of $f(x)=4x-7$.","back":"$f^{-1}(x)=\\frac{x+7}{4}$.","points":2,"mcqOptions":null},{"id":"f25608be-ee17-4a80-958a-c2fecac6b11e","cardType":"basic","front":"How can two proposed inverse functions be verified by composition?","back":"Show that both compositions simplify to $x$.","points":2,"mcqOptions":null},{"id":"a2b5b874-c965-444a-8484-bf845eecc933","cardType":"cloze","front":"The domain of $f^{-1}$ is the [ ].","back":"The domain of $f^{-1}$ is the range of $f$

3.1.5 Algebra and functions (A-level only) Flashcards

3.1.5 Algebra and functions (A-level only) Flashcards