Questions

1. f : R → R and f(x) = 4x³ + 7, check if it is bijective?
2. State whether the function is one-one, onto or bijective?
   • f:R→R defined by f(x) = 3-4x
   • f:R→R defined by f(x) = 1+x²
3. Let A = R-{3} B = R-{1} if f:A→A be defined by f(x)=(x-2)/(x-3), x∈A then show f is bijective.
4. Show that f:N→N, given by f(x) = 2x is not bijective?
5. Show that f:N→N, given by f(1) = f(2) = 1 and f(x) = x-1 for every x>2 is onto but not one-one.
6. f:R→R be defined as f(x) = x⁴ check one-one & onto.
7. Show that function f:N→N given by f(x) = { x+1, if x is odd and x-1, if x is even } is bijective
8. f : R_*→R_* f(x) = 1/x show that function is one-one & onto where R_* is the set of all non-zero real number.
9. Check the injectivity and surjectivity of the following functions:
   • f : N→N given by f(x) = x²
   • f : Z→Z given by f(x) = x²
   • f : R→R given by f(x) = x²
   • f : N→N given by f(x) = x³
   • f : Z→Z given by f(x) = x³
10. f:{2,3,4,5} → {3,4,5,9} and g:{3,4,5,9} → {7,11,15} f(2)=3, f(3)=4, f(4)=f(5)=5 & g(3)=g(4)=7, g(5)=g(9)=11 find gof
11. Find gof and fog is f:R→R and g:R→R are given by f(x)=cosx and g(x)=3x² show that gof≠fog
12. show that f:R-{7/5} → R-{3/5} is defined by f(x)=(3x+4)/(5x-7) and g:R-{3/5} → R-{7/5} if defined by g(x)=(7x+4)/(5x-3), then fog=I_A and gof=I_B where A=R-{3/5} & B=R-{7/5}: I_A(x)=x, ∀ x ∈ A and I_B(x)=x, ∀ x ∈ B are called identify function on sets A and B.
13. Show that if f : A→B and g : B→C are one-one then gof : A → C is also one-one.
14. Show that if f : A→B and g : B→C are onto, then gof : A→C is also onto.
15. Let f : N→Y be a function defined as f(x)=4x+3, where Y={y∈N : y = 4x+3} show that f is invertible and find the inverse.
16. Let f : {1,3,4} → {1,2,5} and
    g : {1,2,5} → {1,3}
    f={(1,2),(3,5),(4,1)} & g={(1,3),(2,3),(5,1)} find gof
17. find gof and fog
    • if f(x)=|x| and g(x) = |5x-2|
    • f(x)=8x³ and g(x) = x^1/3
18. If f(x) = (4x+3)/(6x-4) , x ≠2/3 show that fof(x)=x for all x≠2/3. What is inverse of f?
19. State with reason whether the following function have inverse or not.
    • f:{1,2,3,4} → {10}
      f={(1,10),(2,10),(3,10,(4,10))}
20. f:[-1,1]→R, prove that f(x)=x/x+2 is one-one & find inverse
21. f : R₊ → [4,∞)
    f(x) = x²+4
    so that f^-1(y) = (y-4)^1/2
22. Let f : R→R be defined as f(x) = 10x+7. Find the function g:R→R such that gof=fog=I_R

• Let R = {(1,2), (3,4), (2,2)} and S = {(4,2), (2,5), (3,1), (1,3)} . Find the compositions RoS, SoR, Ro(SoR), (RoS)oR and RoRoR.
• Define the inverse function. Find the inverse of the function f : R → R defined by f(x) = 3x + 2 .
• Show that the function f : R → R defined by f(x) = x² is one-to-one but not onto.
• Let f : R → R and g : R → R be function defined by f(x) = sin x and g(x) = x², find fog and gof .
• Let f : R⁺→R⁺ and g: R⁺→R⁺ be function defined by f(x) = x and g(x) = 3x + 1 for all x ∈ R⁺ , find fog and gof .
• Let f : Z → Z be a function defined by f(x) = x + 5 . Determine whether the function is invertible or not. If it is invertible, then find its inverse.
• Let f(x) = x + 3, g(x) = x + 4 and h(x) = 2x , find fog(x), hof(x) and gofoh(x) .
• If R is the set of real numbers, then discuss the type of function defined by f : R → R such that f(x) = x² ∀ x ∈ R .
• Obtain the recursion definition for the function f(x) = 5 x and then find f(5).