MATH221 Mathematics for Computer ScienceTutorial Sheet Week 11 – Autumn 20211. (i) Let f : [0; 1) ! R be defined by f(x) = x2 + 1. Show that f is one-to-one but not onto. How can the rangeof f be changed to allow the inverse function f-1 to be defined?(ii) Let f : R ! [0; 1) be defined by f(x) = x4. Show that f is onto but not one-to-one. How can the domain off be changed to allow the inverse function f-1 to be defined?2. (i) Let f : R ! R be defined by f(x) := x3 for x 2 R. Can the inverse function be defined? If so, what is it?(ii) Let f : (0; 1) ! (0; 1) be defined by f(x) := x1 – xfor x 2 (0; 1). The function f is one-to-one and onto; showthat g : (0; 1) ! (0; 1), defined by g(x) := xx + 1for x > 0 is the inverse of f.(iii) The function f(x) = cos x : R ! R is not one-to-one nor onto. How can the domain of f be changed to allowthe inverse function f-1(x) = arccos x to be defined?(iv)The function f(x) = tan x : R ! R is not one-to-one but is onto. How can the domain of f be changed toallow the inverse function f-1(x) = arctan x to be defined?(v) The function f(x) = ex : R ! R is not onto. How can the range of f be changed to allow the inverse functionf-1(x) = ln x to be defined?3. Let G = (V; E) be the graph given below.Draw at least two subgraphs H = (VH; EH) of G for which Pv2VH δ(v) = 4.4. In each case below, draw a graph with the specified properties (several answers may be possible).(i) A graph with four vertices of respective degrees 1, 2, 3 and 4.(ii) A graph without loops or parallel edges in which each vertex has degree 3 and which has exactly 6 edges.(iii) A graph without loops or parallel edges with four vertices of respective degrees 1, 1, 2 and 2.(iv) A simple graph with five vertices of respective degrees 2, 3, 3, 3 and 5.(v) A simple graph with five edges and with four vertices of respective degrees 1, 1, 3 and 3.(vi) A graph with four vertices of respective degrees 1, 1, 2 and 6.5. By suitably labelling the vertices (and, if necessary, the edges) of the two graphs below, show that the graphsare isomorphic.

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