Let S be the set (-2, -1, 0. ..., 7, 8) (i.e. the integers from -2...

  
 
 

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Let S be the set (-2, -1, 0. ..., 7, 8) (i.e. the integers from -2 to 8). (a) In this question you will define functions by specifying the output for each potential input. There is no need to create a mathematical expression or formula for your functions. (ii) (i) Define a function f from S to S which is both one-to-one and onto, but which is not the identity function (i.e. f(x)=x for all x in S). Explain how you know it is one-to-one and onto. [2] -2 -1 0 1 2 3 4 5 6 7 8 X f(x) Define a function g from S to S which is neither one-to-one nor onto. Explain how you know it is neither one-to-one nor onto. [2] -2 -1 0 1 2 3 5 6 7 8 X g(x) 4 (b) **(i) Define a function h from S to S with the property that h(x) = h'(x) for all x in S, but which is not the identity function (i.e. h(x)*x for all x in S). Explain how you created this function. -2 -1 01 2 X € h(x) h(h(-1)) = 3 4 5 6 () f(h¹(6))= 7 [2] N Using the functions f.g, and h from (a), find the values of the following expressions, showing all the intermediate steps you took to get your answers. (1) f(h(g(8))) = [1] 8 [1] E [1] (C) Define a (non-trivial) function w(x) using an algebraic expression. Your function should include at least one squaring operation and at least one addition by a number. For example, w(x) = (x + 2)² or w(x) = x² + 1.Your function needs to create a valid output for the inputs x = -1,0,1,2. ** Bonus marks are available in this question if your expression contains a division as well as satisfying the other conditions. You should not choose an example from lectures or workshops. w(x) = (1) (11) If the domain of wis (-1,0,1,2), what is the range of w? Range(w) = Find an algebraic expression for w-¹(x). W-¹(x) = [1] [2/3] **(ii) Would your formula for the inverse function be valid if the domain and range of w were extended to include all real numbers? Explain your [2] answer. Let S be the set (-2, -1, 0. ..., 7, 8) (i.e. the integers from -2 to 8). (a) In this question you will define functions by specifying the output for each potential input. There is no need to create a mathematical expression or formula for your functions. (ii) (i) Define a function f from S to S which is both one-to-one and onto, but which is not the identity function (i.e. f(x)=x for all x in S). Explain how you know it is one-to-one and onto. [2] -2 -1 0 1 2 3 4 5 6 7 8 X f(x) Define a function g from S to S which is neither one-to-one nor onto. Explain how you know it is neither one-to-one nor onto. [2] -2 -1 0 1 2 3 5 6 7 8 X g(x) 4 (b) **(i) Define a function h from S to S with the property that h(x) = h'(x) for all x in S, but which is not the identity function (i.e. h(x)*x for all x in S). Explain how you created this function. -2 -1 01 2 X € h(x) h(h(-1)) = 3 4 5 6 () f(h¹(6))= 7 [2] N Using the functions f.g, and h from (a), find the values of the following expressions, showing all the intermediate steps you took to get your answers. (1) f(h(g(8))) = [1] 8 [1] E [1] (C) Define a (non-trivial) function w(x) using an algebraic expression. Your function should include at least one squaring operation and at least one addition by a number. For example, w(x) = (x + 2)² or w(x) = x² + 1.Your function needs to create a valid output for the inputs x = -1,0,1,2. ** Bonus marks are available in this question if your expression contains a division as well as satisfying the other conditions. You should not choose an example from lectures or workshops. w(x) = (1) (11) If the domain of wis (-1,0,1,2), what is the range of w? Range(w) = Find an algebraic expression for w-¹(x). W-¹(x) = [1] [2/3] **(ii) Would your formula for the inverse function be valid if the domain and range of w were extended to include all real numbers? Explain your [2] answer.