`Hey,
Note: Brother if you have any queries related the answer please do comment. I would be very happy to resolve all your queries.
1)
f(x)=2x+1 is one to one and also onto since it is monotonic and the inputs are real number
2)
f(x)=x^2+1 is not bijection since it gives same output for x=1 and x=-1
3)
f(x)=x^3 is a bijection since it is monotonic in nature and will output all real numbers for real number input
4)
f(x)=(x^2+1)/(x^2+2) is not a bijection since it is not onto because (x^2+1)<(x^2+2). So, 0<=f(x)<1. So, it doen't cover full range hence it is not bijection.
Kindly revert for any queries
Thanks.
(2) [12pts] Argue whether or not the following functions, from R to R, are bijections 1....
Determine whether the following set of functions on R is linearly independent: {1 + x, x + x 2 , x2 + x 3 , x3 + 1} .
Find the derivative of the following functions: (x2-1) f(x) = (x2 +1) f(x) = (x3 + 2x)3(4x + 5)2
5. Determine whether each of these functions is a bijection from R to R. (a) /()=2x-10 (b)/(a) - 4 +4 (c)/(w) - (x + 1)/(x+2) (d)/(a) 7+1
2 -2 3 stem of ODES, f = | ,has the following field: (12pts) a Calculate the value of the slope plotted at the coordinate (2,1) didn't use the equations: x 2x1-2x2= Ax x3 4x -x2 = Ay So, m Ay/Ax Q3B 1 b. On the slope field to the right, plot the trajectory of the solution to the system of ODES having the initial condition: DO NOT SOLVE THE SYSTEM
2 -2 3 stem of ODES, f = |...
Compute the inverse function of each of the following bijections. a. f: R → R,f(x) 4x + 7 ,b,f: (0,oo) → R,f(x)-log8x + 5 c. f: R → R,f(x)--7(x-2)3 + 11, d. f: RM0)-A(0), f(x) = x
(1) For each of the following functions, determine if it is injective and determine if it is surjective. Justify your answer. (a) f : R → R, f(x) = 2x + 3. (b) g : R → R 2 , g(x) = (2x, 3x −1). (c) h : R 2 → R, h((x, y)) = x + y + 1. (d) j : {1, 2, 3} → {4, 5, 6}, j(1) = 5, j(2) = 4, j(3) = 6. (2)...
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For each of the following functions, state whether or not the function is one-to-one, onto, both, or neither: 1) f : Z → Z defined by f(x)=2x + 1; 2) f : R → R defined by f(x)=2x + 1;
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Find the gradient and hessian of each of the following functions and use the hessians to check whether the functions are convex. (b) f:R2 + R given by f(x,y) = x3 – 2xy - y6. (c) f: R3 + R given by f(x1, x2, x3) = cos(x1) + 222} eigenvalues of the hessian). (you may use a computer to find the