Give an example of two functions f : R → R and g : R → R such that f is inj (go f) : R → R is not.

Q: Prove that there does not exist any function f(x, y) such that

A: To show that there does not exist any function that satisfies the given conditions

Q: Let the function f: R→ R be defined by f(x) = { Question 1 0, x2, x=0, x = 0. (a) Explain the…

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Q: Prove that the function f : R → R, given by f(x) = 2x, is one-one and onto.

A: Given, function f : R → R, given by f(x) = 2x.

Q: Let f: A→ X X Y be defined as f (a) = (f₁ (a). f2 (a)) each a ≤ A, Then f is continuous if and only…

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Q: Prove that f: R R where f (x) = |x| is neither injective surjective. nor

A: To prove this theorem, we use the defination of one one and onto.

Q: Let f: R - R be defined by f(x) = |x+2, r20 interval ]1, 3[ under f) is (a) ]3, 5[U] – 1, 1[ (b) (0.…

A: given that,  f(x)=x-2, x<0x+2, x≥0

Q: Show that if f , g, and h are differentiable, then                                          [f…

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Q: Let f be a differentiable function. Consider the relation = x•f where z is implicitly defined in…

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Q: Assume that f is an odd function defined on R and t e Rt. Compute

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Q: Show that the function f : R → R,defined as f(x) = x2 , is neither one-one nor onto.

A: We will show this by counter examples.Let x= 2 and -2 Then f(2)= 2^2 = 4 and f(-2)= (-2)^2 = 4 This…

Q: Use the Product Rule twice to prove that if f, g, and h are differentiable functions of x, then

A: Let us assume f(x)*g(x) together. and h(x) separate.  Then apply the product rule

Q: Use the product rule to differentiate the function h, where a and b are real numbers. h(x) = e¯ax…

A: According to product rule, if f(x)=u(x)*v*x) then f’(x)=u(x)*v’(x)+v(x)*u(x).

Q: Let f: R → Z be defined by f(x) = |x|. The the function f is one to one only onto only one to one…

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Q: Show that the function f(z)= Im(z) is nowhere differentiable.

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Q: Determine whether the function f: R- R, f(x) =,x # 3 is surjective. x+1 %3D r-3

A: Consider the provided question, We need to determine whether the function f:R→R fx=x+1x-3, x≠3 is…

Q: Let f(x, y) = x* /I Evaluate f(3, 0)

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Q: Give an example where | f(x)dg(x) exist even through f is unbounded on [a, b].

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Q: Let f: R" x R → R define as follows f(x,y) = a, then f is

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Q: Let f be differentiable on R. Suppose that f'(r) # 0 for every r. Prove that f has at most one real…

A: Given data is: Let f be differentiable on R. Suppose that f'x≠0 for every x. To prove f has at most…

Q: Let f and g represent differentiable functions such that f ″ ≠ 0 and g″ ≠ 0.

A: Concavity of a function can be determined using the second derivative of the function. If the second…

Q: Prove that if f is continuous, then f(t)(x-t) dt = f(u) du ) dt

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Q: Let f be differentiable on the closed interval [a, b] such that f′(a) = y1 and f′(b) = y2. If d lies…

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Q: Let F (x) = %3D In tdt then F' (x) = et3

A: To find F'(t)

Q: Let f(x, y) = xy - 4x on

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Q: Show in each case that f is differentiable at each a = (a,b) E R²: (a) f : R² → R with f(x, y) = x +…

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Q: Show that the function f : R→ R : f(x) = x° is one-one and onto.

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Q: Let f:R-R defined by f(x) =x² , then f ( x) is bijective صواب ihi

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Q: Differentiate the function.

A: Given f(u)=u1+lnu Now,differentiate function f(x) w.r.t to 'u'…

Q: if f (r) is differentiable prove that f(r)r is irrotational

A: The objective here is to prove that the vector field f(r)r is irrotational. Given: f(r) is…

Q: 4. Define f:R → R, define f (x) = 3x + 5 for all x e R. Show that a. fis one - to - one; b. f maps R…

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Q: Let f be differentiable on R. Suppose that f'(x) # 0 for every r. Prove that f has at most one real…

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Q: Define f: R → [2, 0) by f(x) = x² +2,Vx € R.Prove that f is onto.

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Q: Assume that f : R → R is differentiable everywhere and satisfies for all h, x E R: f(x + h) – f(x) =…

A: Let's prove this.

Q: Let f:R-R defined by f(x) =x² , then f(x) is bijective صواب ihi

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Q: Let f be a function from A to B . Define a relation R by (x,y) ∈ R If and only if  f(x) =f(y) .Prove…

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Q: Define the function f:R → R by f(x) = x³ + 1, Vx € R. a) Prove that f is onto? b) Prove that f is…

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Q: Let f:Z → Z be given by f (x) = 3x + 1. a. Show that f is one-to-one. 7. b. Is f onto? EXPLAIN.

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Q: Let f (x) = vr +2² + 2 , ro = 1, h = 0.5, r1 using the Three-Point formulas to approximate f(1) (1);…

A: Given: The given fx=x3+x2+2, x0=1,h=0.5,x1=1.5 and x2=2

Q: prove that if f(x)is derivable at a point say "a" then it is continous at x=a

A: Here we have given that the function fx is diffrentiable at a point  .x=a that is f'x exists.…

Q: Consider the function f: (-2,0) → [0, 0) defined by f (x) = Vx + 2. Prove that f is suriective.

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Q: Suppose f is a bounded real function on [a, b), and f' eR on [a, b). Does it follow that fe R? Does…

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Q: The Fundamental Theorem of Calculus says that f(x)da = F(a) – F(b) provided f = F' Is continuous on…

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Q: Let F = (2y? + z)i+ 4ryj+ æk. (a) Show whether F is conservative.

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Q: 4. Consider the function f: R → P(R) defined as f(x) = {y ER|y² <x} (a) Find f(4) and write it as an…

A: Given f: ℝ → P(ℝ)  defined as f(x) = {y ∈ ℝ  | y2 ≤ x}

Q: Consider the functions g : R→ R defined by g(x) = x^2+4, x∈R and h : R→R defined by h(x)=x−2 ,x∈R.…

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Q: Consider the function f : [a, b] → (a, b] with ƒ being continuous. Show that there is a number c in…

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Q: Prove that the followings are entire functions: (a) f(z) = z", Vn e Z* %3D

A: As per our company guidelines we are supposed to answer 1st question  . Kindly repost the other in…

Q: Let f: R" x R → R define as follows f(x, y) = then f is

A: Given that: f:Rn×Rn→R define as follows fx,y=∑xiyi¯i=1n. The objective is to determine the value of…

Q: Let the function f: R→ R be defined by { f(x) = 0, 1 x2, x=0, x = 0.

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Q: Consider the function f : Z × Z → Z × Z defined by f(a, b) = (a+ b, a b) -

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