Inverses and Radical Functions

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College AlgebraInverses and Radical FunctionsRestrict the domain to find the inverse of a polynomial functionSc far, we have been able t o find the inverse functions c fcubic functionswithout having to restrict their domains. However, as we know, not allcubit polynomials are one-to-one. Some functions that are not one-to-one may have their domain restricted so that they are one-to-one. butonly over that domain. The function over the restricted domain would then have aninverse function.Since quadratic functions are not one-to-one. we must restrict their domain in order to find their inverses.A General Note: Restricting the DomainI f a function is not one-to-one, it cannot have an inverse. If we restrict the domain of the function so that it becomes one-to-one. thus creating anew function, this new function will have an inverse.How Tc: Given a polynomial function, restrict the domain of = function that is not one-to-one and then find the inverse.Restrict the domain by determining a domain on which the original function is one-to-one.Replace flyl with y.niterchangexand y.Solve fory and rename the function or pair of function/Revise the formula for/by ensuring that the outputs of the inverse function correspond to the restricted domain of the original function.Example 3: Restricting the Domain to Find thenverse o f a Polynomial FunctionFind the inverse function of it/ (*) = ( *4)1, x > 4/ ( r )=(u:4)i, a : < 4SolutionThe original function/ ( « ) = ( »4)4is not one-to-one, but the function is restricted to a domain o fi> 4Figure5

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To find the inverse, start by replacingf (®)with the simple variabley.y= ( x4)Interchange -r andy.x— ( y4)Take the square root..Lii/x — y4A d d 4t o both sides.I4 1y<r-yThis is not a function as written. We need to examine the restrictions o n the domain o f the original function to determine the inverse. Since wereversed the roles of x and y for the original fix), we looked at the domain: the values x could assume. When we reversed the roles o f x ar d y, thisgave us the values y could assume. For this function.z >4, so for the inverse, we should haveJf >4, which is what our inverse function gives.The domain of the original function was restricted t oz>4, so the outputs of the inverse need to be the same,/> 4. and we must use the - case:/* (x) = 4 IThe domain of the original function was restricted toz<4, so the outputs of the inverse need to be the same,/ W < 4, and we must use the - case:/1(x) = 4Analysis of the SolutionOn the graphs below, we see the original function graphed o n the same set of axes as its inverse function. Notice that together the graphs showsymmetry about the line7 = 2 -. The coordinate pair(4,0)is on the graph of f and the coordinate pair( 0 , 4 )is o n the graph of/1. For any coordinate pair, if [a, t j is o n the graph of f then (ir. tr) is on the graph of/1. Finally, observe that the graph o ffintersects the graph of/1o r the line y = x. Points of intersection for the graphs of f and/1

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Example 4: Finding the Inverse of a Quadratic Function When the Restriction Is N o t SpecifiedRestrict the domain and then find the inverse of/ ( » )= ( *2)13SolutionWe can see this is a parabola with vertex at(2,3)that opens upward. Because the graph will be decreasing on one side o f the vertex and increasing o n the other side, we can restrict this functionto a domain o n which it will be one-to-one by limiting the domain t ox > 2To find the inverse, we will use the vertex form of the quadratic. We start by replacing fbj with a simple variable, y, then solve forxy = ( x2 )23lutcrcluiiigi: x andy.X =2)*3A d d 3 to both sides.IXI 3 = ( y2)2Take the square root.1J - / x I 3 =j2Add 2 to b o t h sides.2 1v'xI 3yRename the lurictioti.J1(x) = 2 1Now we need t o determine which case t o use. Because we restricted our original function to a domain o fx >2, the outputs of the inverse should be the same, telling us t o utilize the + case/1( x ) = 2II f the quadratic had not been given i n vertex form, rewriting it into vertex form would he the first step. This way we may easily observe thecoordinates of the vertex to help us restrict the domain.Analysis o f the SolutionNotice that we arbitrarily decided to restrict the domain o nx > 2. We could just have easily opted to restrict the domain o nx <2, i n which case

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