6. Determine the domain and range of each of the following functions and whether they are...

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6. Determine the domain and range of each of the following functions and whether they are one-to-one or onto: (a) f- ((r, 9) €N x Ny = 2r + 3} (b) f= {(1, v) EN x Ny=-4} (c) f= ((r, y) €Zx Z|y--3} 10, 00) and let f be a subset of D x R defined by f= {(r, y) : =y+1). Is f a 7. Let D = real-valued function with domain D? 8. Find the domain and range of each function: (a) /: 10,1)- -R defined by f(z) = (b) f(-) = (e) f: Z R defined by f(r) = |r|.- 9. Let f: R-R be defined by f(r) =1+r. (a) Prove that / is bijective. (b) Find f and verify that fof= l=fof. 10. Show whether or not the function is invertible. If yes, find its inverse. (a) f:RR if defined by f(r) = -1. (b) f: (0, )R if defined by f(r) =-1 (c) f: (-0,0) – -Rif defined by f(x) =- 1. (d) f(z) = defined on its domain. %3D (e) S(r) = V-z defined on its domain. 11. Let f: N-Z be defined by S(2) = +2 and g: N-Z be defined by g(n) =2 n. (a) Determine whether fog is well defined; that is, whether the range of the function g is a subset of the domain of the function f. If it is well defined, write a formula for (fog)(2) explicitely. (b) Determine whether gof is well defined; that is, whether the range of the function f is a subset of the domain of the function g. If it is well defined, write a formula for (go )(x) explicitely. 12. Let /: (1, o) R be defined by f(r)-- 2r+1. (a) Show that f in 1-1, or give a counterexample. (b) Show that / is onto, or give a counterexample. 13. Let /:N- Z be defined lay flr) =-1. (a) Show that /is 1-1, or give a counterexample. (b) Show that / ls onto, or give a counterexample. 14. (a) Let f:D Aand g: A C. Prove that if gof: D l is onto then g is onto (b) Asnume noow that :D- A and g: A C. Prove that if gof: D Cin one-to-one then fin one-to-on. 6. Determine the domain and range of each of the following functions and whether they are one-to-one or onto: (a) f- ((r, 9) €N x Ny = 2r + 3} (b) f= {(1, v) EN x Ny=-4} (c) f= ((r, y) €Zx Z|y--3} 10, 00) and let f be a subset of D x R defined by f= {(r, y) : =y+1). Is f a 7. Let D = real-valued function with domain D? 8. Find the domain and range of each function: (a) /: 10,1)- -R defined by f(z) = (b) f(-) = (e) f: Z R defined by f(r) = |r|.- 9. Let f: R-R be defined by f(r) =1+r. (a) Prove that / is bijective. (b) Find f and verify that fof= l=fof. 10. Show whether or not the function is invertible. If yes, find its inverse. (a) f:RR if defined by f(r) = -1. (b) f: (0, )R if defined by f(r) =-1 (c) f: (-0,0) – -Rif defined by f(x) =- 1. (d) f(z) = defined on its domain. %3D (e) S(r) = V-z defined on its domain. 11. Let f: N-Z be defined by S(2) = +2 and g: N-Z be defined by g(n) =2 n. (a) Determine whether fog is well defined; that is, whether the range of the function g is a subset of the domain of the function f. If it is well defined, write a formula for (fog)(2) explicitely. (b) Determine whether gof is well defined; that is, whether the range of the function f is a subset of the domain of the function g. If it is well defined, write a formula for (go )(x) explicitely. 12. Let /: (1, o) R be defined by f(r)-- 2r+1. (a) Show that f in 1-1, or give a counterexample. (b) Show that / is onto, or give a counterexample. 13. Let /:N- Z be defined lay flr) =-1. (a) Show that /is 1-1, or give a counterexample. (b) Show that / ls onto, or give a counterexample. 14. (a) Let f:D Aand g: A C. Prove that if gof: D l is onto then g is onto (b) Asnume noow that :D- A and g: A C. Prove that if gof: D Cin one-to-one then fin one-to-on.