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Let f: B → A be given by f(x) = (3x + 4)/(5x − 7) and g: A → B be given by g(x) = (7x + 4)/(5x − 3), where A = R {3/5} and B = R {7/5}. If I_A and I_B denote the identity maps on A and B respectively, then which statement is true?

1. f∘g = I_A and g∘f = I_A
2. f∘g = I_A and g∘f = I_B
3. f∘g = I_B and g∘f = I_B
4. f∘g = I_B and g∘f = I_A

Correct answer: f∘g = I_A and g∘f = I_B

Solution

Computing f(g(x)) = x and g(f(x)) = x. Since f:B->A and g:A->B, f o g maps A->A so f o g = I_A, and g o f maps B->B so g o f = I_B.

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