Q4) Let functions f, g, h from V = {1,2, 3, 4} into V be defined by: f(n) = 6 – n, g(n) = 3, h = {(1, 2), (2, 3), (3, 4), (4, 1)} . Decide which functions are: (a) One-to-one; (b) onto; (c) both; (d) neither.

Q: 4.) Prove that (F-G)(v)=F(v)-G(v) for all F, GEg (V,W)

A: L(V,W) , a set of linear transformation from V(F) to W(F) , is a vector space over the field F with…

Q: Give Three Functions  of Wronskian ?

A: To Give: Wronskian of three functions.   Wronskian:

Q: 2. Determine whether f is a function from Z to R if a) f(n) =±n. b) f(n) = Vn²+1. c) f(n) = 1/(n² –…

A:

Q: Explain the Wronskian of 2 functions f and g ?

A: Let f and g are the two functions. The Wronskian of the n functions is the determinant of the matrix…

Q: Find f f and f and evaluate each at the given point. f(x, y, 2) xy + 2xyz-Syz, (-3, 1, 3) ,(x, Y, z)…

A:

Q: 8. Is the set of functions f₁(x) = 3e³x and f₂(x) = -3e3³* independent? Show using the -3.x…

A: Given functions- f1x=3e3x  and  f2x=-3e-3x To show whether the given set of functions is independent…

Q: Consider the function d: R² × R² → R defined by min{|y2 - x₂, 1} if x₁ = y₁ otherwise, d(x, y) :=…

A: A metric on a set X is defined as a function d: X×X→R which satisfies the following conditions:…

Q: Suppose that z = f(x, y) is implicitly defined as a function of x and y. Determine for 4x²z³ - 5yz²…

A:

Q: Let f : {1, 2, 3} → {a, b, c} be one-one and onto function given by f(1) = a,   f(2) = b and f(3) =…

A: Consider

Q: Describe the long run behavior of f(z) = z" As z + - 00, f(z) - As z + 0, f(z) → ( -0-

A:

Q: Let f: [2,3] → {1,3} be a continuous function such that the equation f(x) - x = 0 has a solution in…

A:

Q: Let f be the function defined byf (z) = 3z2 -z. What is the absolute minimum value of f on the…

A: Topic = Application of Derivative  The absolute minimum value of f is 0.

Q: Present a combinatorial argument to justify your answer. How many functions are there from {1, · · ·…

A: Consider the given information as, 1,⋯,n to 0,1

Q: Find

A:

Q: 4. Check that the function f(x) = x3 satisfy the hypothesis of mean value theorem or not at the…

A: To check that f(x)=x4/5 satisfies mean value theorem or not on [0,1]

Q: Find all values of c that satisfy the conclusion of Cauchy’s MeanValue Theorem for the given…

A: Consider the following functions: fx=xgx=x2 Differentiate fx and gx with respect to x: f'x=1g'x=2x

Q: 7. has a fixed point. ) Prove that if f is a continuous function from [2, 5] to [2, 5], then f

A:

Q: Show that the function f : [0, 1] →[0, 1] given by f(x) = (x −1)2 has a fixed point. Can you apply…

A:

Q: Select the option that gives a list of the stationary points of the function f(x, y) = x – 6x² – 3xy…

A: we have given f(x, y)  We have to find the stationary points

Q: Suppose that f(2, 3,1) = 5, fz(2,3,1) = 2, fy(2,3,1) = 3 and f.(2, 3,1) = -1. %3D Then f(1.9, 3.3,…

A:

Q: (b). Find the bound for absolute error at x = 2.9 on the interval [x, x,], where x, = 2.25, x, =…

A: On the interval x0,xn where x0=2.25, x1=2.5, x2=2.75, x3=3 with n=3 along with the function…

Q: Let f be a function defined on (- ∞,00). If f"(x) = x² – 5x –6 then the critical numbers of f(x)…

A: Using definition of critical point of function.

Q: Find the number of all onto functions from the set {1, 2, 3, ... , n} to itself.

A: Consider the given set {1, 2, 3,…, n}Since the function is onto. So, all elements of {1, 2, 3,…, n}…

Q: Find four relations from {a, b} to {x, y} that are not functions from {a, b} to {x, y}.

A: It is given that the two sets are {a, b} and {x, y}. It is known that if A and B are the sets then…

Q: 2: A necessary condition for a function to be unimodal on the interval [a, b] is that strictly…

A: Statement: A necessary condition for a function to be a unimodal on the interval [a,b] is that…

Q: 4. Verify Rolle's Theorem for f(x) = x² - 2x +1 on [-1,3]

A: Given the function f(x)=x2−2x+1 On the interval [−1,3]

Q: Differentiate the function.

A: Given function -f(u) = lnu1+ln(2u)Differentiating with respect to u - f'(u) = ddulnu1+ln2uf'(u) =…

Q: (a) What is the number of onto functions f : {1,2, 3, 4, 5} → {1,2, 3}? (b) What is the number of…

A:

Q: The domain of the function g(t) = vt – 5 is O R-{5} O R O (5, O [5, 0) о (-оо, —5) (5, 00)

A: From the given function: gt=t-5

Q: If the graph of a linear function z = f(x, y) travels through the points (1, 2, 3), (2, 2, 4) and…

A: here option (a) is true because this is a linear function if graph of this function passes through…

Q: Let f(z) = 12 -エ %3D f(z) = f(2)

A: Given,          fx=12-x

Q: What is the value (i.e. what do we learn) from using the Wronskian on a set of functions?

A: Let F = { f1, f2, f3, ...fn} be the set of functions. 1. If W(f1, f2, f3,...fn) =0 Then we say…

Q: If f(x,y)=x²+3xy-2x+3, then f(x+h,y) is... Jika f(x,y)=x2+3xy-2x+3, maka f(x+h,y) adalah... 1. O…

A:

Q: 5. Let f, and f, be the function from R to R such that f,(x) = x²+2x+3 and f,(x) = x-x. What are the…

A:

Q: For the function def ined by f (x) =x² on [0, 1] and P=0,-,- 4 4 4 1, then: U(f;P)=[X]

A: We will find U(f,P) for the partition

Q: 2. Consider the function а(т — 2) Vx +2 – 2 - if x 2, 2x 2 x² - where a, b, c are real numbers.…

A: Given function is  f(x)=a(x-2)x+2-2   if x<2b                     if x=21x-2-cx2-2x  if x>2…

Q: Let f(x, y) = x²y - e. Evaluate f,(2, -3) and f,(7, In(5)). f(2, -3) = f,(7, In(5)) =

A:

Q: Find a function f that satisfies the conditions. "(a) --Na (1) - 5, (4) - -1

A: Given Data: The second derivative of the function is: f"x=x-3/2 ......1 The value of the first…

Q: 21. The interval for function 13y-71 4 is (1,11/3) O[1,11/3] (-00,1)u(00,11/3) 0[1,11/3)

A:

Q: Evaluate the function at the specified points. f(x, y) = y+ xy', (-2,4), (1, –3), (3, 1) At (-2, 4):…

A: Topic: function

Q: 2- Let f(x) = ()*O- , x = 0,1, is p.d.f then P(x = 0) =

A: The p.d.f.  of random variable X is: f(x)=13x231-x ;    x=0, 1

Q: Find the values of m and n that make f(x) continuous over (-∞, ∞): f(x) = m= n= 4 x-2 mx² - nx 2x -…

A: Topic- Limits and continuity

Q: [3.9] Let fi, f2,., fk: R" → R be convex functions. Consider the function f defined by f(x) =…

A:

Q: let f(x)={x^2+2, x >-1 AND -x^2-x, x less than for equal to -1} compute f(0), f(-1)and f(-2) a. 0,…

A:

Q: Verify the Mean value Theorem for the function f(r) = 2z² – 8z + 6 on [1.3] %3D

A:

Q: Verify Rolle's theorem for f(x) = x² - 9x +6 over the interval [0, 3].

A:

Q: If z(m) denotes the factorial function of order n then show that A"z(m) = n!h" where h is difference…

A: Use the definitions of forward difference operator ∆fx=fx+h-fx and the  nth factorial of x, to find…

Q: Let f be the function from (a, b, c, d) to (1, 2, 3, 4) with f (a) = 4, f (b) = 2, f (c) = 1, and f…

A: Consider a function f:a,b,c,d→1,2,3,4 with fa=4, fb=2, fc=1 and fd=3. A function is a bijection if…

Q: Find all the points of discontinuity of the function S(x) = 1- x    0 < x ≤ 1      X + 2     1< x <…

A: