Define H: R XR RxR as follows. H(x, y) = (x + 8, 9 - y) for every ordered pair (x, y) ER × R (a) Is H one-to-one? To answer this question, suppose (X1, Y₁) and (X2, y2) are in Rx R and H(X1,Y1) = H(X2 Y2). Apply the definition of H to both sides of this equation. (x + 8, 9 − y) = ( ) - Then apply the definition of ordered pair to the result. ×₁₁ + 8 = and 9 1 - = Solve these equations and apply the definition of ordered pair to obtain the result that (X₁, Y₁) ? V (×21 Y2). Therefore, H ---Select--- ✓ one-to-one. (b) Is H onto? Let (u, v) be any ordered pair in R x R. On a separate piece of paper, solve the equation (u, v) = (x + 8, 9 − y) to find values for x and y. Use the result to fill in the ordered pair below. (x, y) = (i) Is (x, y) in R x R? The numbers x and y ---Select--- real numbers because differences of real numbers ---Select--- ✓ real numbers. Hence, (x, y) ---Select--- in R x R. (ii) Does H(x, y) = (u, v)? According to the formula that defines H, we get the following in terms of u and v. H(x, y) = H + 8,9 - (u, v) (iii) Conclusion: H -Select--- onto.

Define H: R XR RxR as follows. H(x, y) = (x + 8, 9 - y) for every ordered pair (x, y) ER × R (a) Is H one-to-one? To answer this question, suppose (X1, Y₁) and (X2, y2) are in Rx R and H(X1,Y1) = H(X2 Y2). Apply the definition of H to both sides of this equation. (x + 8, 9 − y) = ( ) - Then apply the definition of ordered pair to the result. ×₁₁ + 8 = and 9 1 - = Solve these equations and apply the definition of ordered pair to obtain the result that (X₁, Y₁) ? V (×21 Y2). Therefore, H ---Select--- ✓ one-to-one. (b) Is H onto? Let (u, v) be any ordered pair in R x R. On a separate piece of paper, solve the equation (u, v) = (x + 8, 9 − y) to find values for x and y. Use the result to fill in the ordered pair below. (x, y) = (i) Is (x, y) in R x R? The numbers x and y ---Select--- real numbers because differences of real numbers ---Select--- ✓ real numbers. Hence, (x, y) ---Select--- in R x R. (ii) Does H(x, y) = (u, v)? According to the formula that defines H, we get the following in terms of u and v. H(x, y) = H + 8,9 - (u, v) (iii) Conclusion: H -Select--- onto.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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