Consider a function f : R → R given by f(x) = 2x - |x| + 3. 1. Show that f is injective; 2. Show that f is surjective.

Q: 2. Prove that the function f : R? → IR defined by f(x, y) = 1- a2 – 4y? is neither injective nor…

A: Given the function f:ℝ2→ℝ defined by fx,y=1-x2-4y2

Q: Let the function ƒ : R → R be given by 1- cos(x) if x + 0, f(x) := if x = 0. %3D 1 Prove that f is…

A: The solution is given as

Q: 5. Let f : R → R be defined by f(x) = { 0 if r € Q, x2 otherwise. Prove that f is continuous at 0…

A: Here we prove the problem using the definition of continuity and a theorem.

Q: f(x, y) = x² + y² +1 f (x, y)

A: Let's find.

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

A:

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 : R → (−1, 1) be given by f(x) = x/(1 + |x|). Show that f is differentiable at x = 0, with…

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Q: (b) Test the continuity of the function f(x) = sgn(]x|) at x = 0. Is f(x) differentiable at x = 0?…

A:

Q: (b) Consider the functions g : R → R defined by g(x) = r² + 4, x € R and h : R → R defined by h(x) =…

A:

Q: sketch f(x)=sec x over the half-open interval -r,27). 2 1 3/% 27t 52

A: Given query is to find sketch the graph of sec(x) .

Q: Let h: R → R be a continuous function on all of R. If h(x) T, prove that h(T)<0.

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Q: Prove that the function f(x) = 2x is uniformly continuous on R.

A:

Q: Assume that the function f is such that f(3) = 1, ſ f(x) dx = 5, and f'(x) is continuous on [0, 3].…

A:

Q: Explain why f (x) = sin (x² + 3r + 1) is a continuous function for all c € R. %3D

A: Let us assume that: g(x)=sin(x)and h(x)=(x2+3x+1)So,…

Q: (Q4) Prove directly from the definition that the function f: R→ R defined by f(x) = 3r² +r for all r…

A: Sol

Q: Show by definition that f(x) = |x| : R → R is continuous on R. 2.

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Q: If f : R+ → R is a function such that f(x + y) = f(x)f(y) for all x,y ∈ R+. Prove that f is…

A:

Q: Let h: R → R be a continuous function on all of R. If h(x) T , prove that h(T)<0.

A: Given that, let h:ℝ→ℝ be a continuous function on all of ℝ. Also given, hx≤0 for all x&gt;π.

Q: Let f : R2 → R with f(x,y) = px^2 + y^2. Sketch the graph of f. Prove that f is not differentiable at…

A: Given f(x,y)=px2+y2

Q: Let f : R → R with f(x) = 3x² + 2. Prove that f is differentiable at the point –1 %3D

A: f(x) =3x2+2 f'(x) =limh→0f(-1+h) -f(-1)h        =limh→03(-1+h)2+2-3-2h        =limh→03+3h2-6h -3h…

Q: (a) Let x2 if x is rational f (x) = 0 otherwise. Prove that f is differentiable at 0.

A:

Q: Prove that if f : V → W is differentiable at a point a ∈ V it must be continuous there.

A: I have proved limiting value is equal to functional value

Q: Let f be a continuous function on R such that f(x+y)=f(x)+(y) ¥x, yER. Show that f(x)3cx for some…

A:

Q: 2. Determine if a function f : [0, 27] → R defined by f(x) = sin 2n is continuous.

A:

Q: Let f: RR be a function satisfying f(x + y) = f(x) + f(y) for all x, ye R. If the function f(x) is…

A: Note :   Let r&gt;0 and c∈R ,then vr(c)={x∈R : x-c&lt;r }=(c-r ,c+r) ,which is called r…

Q: Let f:Z x Z+ Z x Z be the function given by f(z, y)= (2z + y,I- y). Find, with proof, an element af…

A:

Q: Let f be the function defined by: sin x, x 2 - For what values of x is f NOT continuous?

A: Given function is: fx=sinxx&lt;0x20≤x&lt;12−x1≤x&lt;2x−3x≥2

Q: Let the function f: R R be differentiable at x = 0. Prove that lim %3D = 0.

A: Given the function f:R→R be differentiable function at x=0. We have to prove that limx→0fx2−f0x=0.…

Q: Consider the function g:Z×Z→Z×Z given by g(x,y)=(x+3y,2x−y). Is g injective? Proof or…

A:

Q: Consider a surjective function f : A → A Prove that fo f o f is surjective

A:

Q: Consider a function f : R→ R given by f(x) =2x · |æ| + 3. 1. Show that f is injective; 2. Show that…

A:

Q: show that f,given by f(x) x +0 )is continuous on R-{0}

A:

Q: Then, write a function f(z) such that a) it is analytic in C except its isolated singular points,…

A:

Q: Let f(2)=u+iv be an analytic function. Given that u=v² show that f(z) is a constant.

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Q: Without the aid of Wrongskian, determine whether the given set of functions is linearly independent…

A:

Q: Let f be a continuous function with f(x) > 0 for all æ € R. Define r(z²+1)" g(r) = e" f(t)dt.

A: Let f be a continuous function with fx&gt;0 ∀x∈ℝ. A function g is defined by  gx=∫0x2+1xexftdt…

Q: Consider the function g : Z × Z →Z × Z given by g(x, y) = (x + 3y, 2x – y). Is g injective? Proof or…

A: The given function is  g : ℤ×ℤ→ ℤ×ℤ given by g(x,y)=(x+3y, 2x-y).

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.

A:

Q: Suppose f is a continuous on R and f(x + y) = f(x) + f(y) for all r,y R. Prove that for some…

A: What is Continuous Function: In mathematics, a continuous function is one that, in response to a…

Q: For the function f(x, y) = xy((x2 − y2) /( x2 + y2) ) define f(0, 0) such that f is continuous at…

A: A function f is continuous at the origin, the function value at the origin must be equal to the…

Q: (1- a? f(x) = 1– x² if x E Q, if r ¢ Q. %3D p at which f is continuous. Prove your conclusion.

A:

Q: Give a 8, e proof that f(x) = x² + 3x is continuous at x = a, where a is any real %3D number.

A:

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: Give an example of a differentiable function g: R→ R such that f is strictly increasing on R but g'…

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Q: Suppose that f(x)=xu(x), where u(x) is a differentiable function at x=2 such that u(2)=3, u'(2)=1.…

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Q: Prove f (x) 1+ mx is continuous at x = 1, where m e R

A:

Q: 16. Consider the function f:(-2, 0) → [0, 0) defined by f (x) = vx +2. Prove that f is injective.

A: f(x) here is injective if and only if for all a,b∈(-2, ∞) we have a=b⇔f(a)=f(b)

Unitary Method

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products. 

Speed, Time, and Distance

Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.

Profit and Loss

The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.

Units and Measurements

Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.