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(a) Expand 3 1 + 4x in ascending powers of x up to and including the term in x 2.  \n(b) By putting x = 0.01 , determine an approximation for 130 .  \n3  \nNotes  \nA mapping is a rule that takes a given set of inputs and assigns each one of them one or more output values using a mapping rule.  \nHere are three types of mappings:  \nOne-to-one mapping  \nEach x−value maps toa unique y−value.  \nE.g. y = x + 1  \nMany-to-one mapping  \nMore than one x−value  \nmaps to the same y−value. E.g. y = x 2  \nOne-to-many mapping  \nOne x−value maps to  \nmore than one y−value. E.g. y 2 = x  \nN.B. With many-to-one mappings, it is enough that one x−value maps to more than one y − value. Similarly for one-to-many mappings.  \nWhen is a mapping a function?  \nA mapping is a function when it is a one-to-one or a many-to-one mapping.  \nThe vertical line test is used to determine whether a mapping is a function.  \nIf any vertical line meets the graph at most once, then the mapping is a function.  \nThere only needs to be one vertical line that meets the graph more than once for a mapping not tobe a function.  \ny = x + 1  \nTo the left all the vertical lines intersect the graph of y = x + 1 once so it is a function.  \nTo the right, at least one vertical line intersects the graph of y 2 = x in at least two places so it is not a function.  \n2 y = x  \n[www.mathspanda.com](www.mathspanda.com)  \nWhen is a function a one-to-one function?  \nThe distinction between one-to-one and many-to-one functions is an important one because only one-to-one functions have inverses. Inverse functions will be discussed in a later lesson.  \nThe horizontal line test is used to determine whether a function is one-to-one or not.  \nIf any horizontal line meets the graph at most once, then the function is a one-to-one function.  \ny = ln x  \nTo the left all the horizontal lines intersect the graph of y = ln x once so it is a one-to-one function.  \nTo the right, at least one horizontal line intersects the graph of y = x 2 in at least two places so it is not a one-to-one function.  \n2 y = x  \nN.B. Only one horizontal line has to meet the graph in more than one place for a function not tobe a one-to-one function.  \nE.g. 1 For the following graphs, determine whether the mapping is a function or not. For those that are functions, determine whether it is a one-to-one function and hence state whether it has an inverse.  \n(a) (b) (c)  \nWorking: (a) Any vertical line meets y = ex only once so it is a function.  \nAny horizontal line also meets y = ex only once so it is one-toone function.  \nHence, y = ex does have an inverse.  \nVideo:  Function notation  \nVideo:  Vertical line test  \nVideo:  Horizontal line test  \nSolutions to Starter and E.g.s  \n[www.mathspanda.com](www.mathspanda.com)  \nExercise  \np16 2A Qu 1-2  \nSummary  \nA mapping is a function when it is a one-to-one or a many-to-one mapping.  \nThe vertical line test is used to determine whether a mapping is a function.  \nIf any vertical line meets the graph at most once, then the mapping is a function.  \nOnly one-to-one functions have inverses.  \nThe horizontal line test is used to determine whether a function is one-to-one or not.  \nIf any horizontal line meets the graph at most once, then the function is a one-to-one function.","cbCaimjJvI99Z2Zs","https://ap.wps.com/l/cbCaimjJvI99Z2Zs","pdf",4233680,"English","# Starter\n## Types of mappings\n## When is a mapping a function?\n## When is a function a one-to-one function?\n# Exercise\n## Summary","[{\"question\":\"What are the three types of mappings described in the lesson?\",\"answer\":\"The lesson lists one-to-one, many-to-one, and one-to-many mappings, each described by how input x-values relate to output y-values.\"},{\"question\":\"How do you tell whether a mapping is a function?\",\"answer\":\"Use the vertical line test: if every vertical line meets the graph at most once, the mapping is a function.\"},{\"question\":\"How do you determine whether a function is one-to-one and has an inverse?\",\"answer\":\"Use the horizontal line test: if every horizontal line meets the graph at most once, the function is one-to-one and therefore has an inverse.\"}]","Mappings and functions - Starter - Lesson notes and exercises | PDF"]