2.26 please!! Definition 2.24.(a) If S is a set and f is a function, then the preimage of S under f is the set{x € domain(f) | f(x) E S}in set-builder notation. We denote the preimage of S under f by f-1(S).(b) If D is a set and f is a function, then the image ofD under f is the set{y € codomain(f) | y = f(x) for some x E D}.We denote the image of D under f by f(D).Exercise 2.25. Suppose f and g are both functions.(a) Determine exactly when the preimage of a set under f is empty.(b) Determine exactly when the image of a set under f is empty.(c) Show, if A and B are both subsets of the codomain of f, then f-'(AUB) = f¯-'(A)Uƒ-1(B).(d) Show, if A and B are both subsets of the codomain of f, then f-'(An B) = f-1(A)n f-1(B).(e) Show, if A and B are both subsets of the domain of f, then f(AUB) = f(A) u ƒ(B).(f) Show, if A and B are both subsets of the domain of f, then f(An B) = f(A) n f(B).(g) Explain why, if C is a subset of the codomain of f and A is a subset of C, then f-1(C\ A) =f-'(C) \ f-1(A).27(h) Explain why, if C is a subset of the domain of f and A is a subset of C, then f(C) \ f(A) is asubset of f(C \ A).(i) Give an example of a function f, a subset of its domain C and a subset A of C for whichf(C\ A) is not a subset of f(C) \ f(A).(i) Explain why f-1(codomain(f)) = domain(f).(k) Explain why f(domain(f)) = range(f).(1) If A is a subset of the domain of f, explain why f-1(f(A)) = A.(m) If A is a subset of the range of f, explain why f(f(A)) = A.(n) If A is a subset of the codomain of f, explain why f(f-1(A)) is a subset of A.(o) Give an example of a function f and subset of its codomain A for which A is not a subset off(f-(A)).(p) Find the domain of fog in terms of the domain of f and g as a preimage.(g) Show, if A is any subset of the codomain of g, then (f o g)-'(A) = g-1(f-'(A)).(r) Show, if A is any subset of the domain of g, then (f o g)(A) = f(g(A)).Exercise 2.26. If f, 9,1, m are functions for which f Ug and lUm are functions, find the domainof (f Ug) o (1 U m) as a union of preimages.

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f, g, l, m, f∪g and l∪mare functions.

Exercise 2.26. If f, g,l, m are functions for which f Ug and lUm are functions, find the domain of (f U g) o (l Um) as a union of preimages.