The function f : R - {2} → R, is defined by fx) = is x-2 A surjective --B- injective C both injective and surjective D neither surjective nor injective

Q: Consider the function f with domain (-2, x): 3|r2 – | if – 2<I<1 2.r2 + 2.r 4 f(1) = 3 if r = 1…

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Q: Let f(x) be defined for a ≤ x ≤ b. Assuming appropriate properties of continuity and derivability,…

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Q: 5. Determine whether the function f : R+ → R defined by the rule f (x) = positive real numbers x in…

A: A function is said to be bijective if the function is both one one and onto. The given function is…

Q: Find the interval that the fullowing function is continous at In (9 -x) %3D A. [I,3] BL) (3,3) (1,3)…

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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: Use the Squeeze Theorem to show that f(x) = x² is in R[0, 3]. %3D

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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 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: R (-1, 1) be a function defined by Prove that (a) f is injective; (b) f is surjective. f(x) =…

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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 the function f: R –→ R defined by f (x) = 3x – 5 is bijective.

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Q: Suppose f, g:C → C are differentiable. Prove that fg is differentiable and that (fg)' = f'g + fg'.

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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: I R not be continuous, where I is a closed interval in R. With appropriate reasoning, is…

A: If a function f is not continuous, then the assumption that  f(b)f(b) has opposite signs doesn't…

Q: Let f : I → R not be continuous, where I is a closed interval in R. With appropriate reasoning, is…

A: Given that f : I → R not be continuous, where I is a closed interval in R.

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: Let f(x) = Vx. Compute the difference quotient for f at a = 7 with h = 4. %3D %3D

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Q: Consider the function h : Z xZ → Z× Z defined by h(x, у) 3D (2ху, 7x — Зу). (a) Is h injective? (b)…

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Q: Find and sketch the domain of f(r, y) In(r - y) 6. -

A: Given:      f(x,y)=lnx-y9-x2-y2·x2+y2-4    Interval notation : -3,0∪0,3   Set builder notation:…

Q: Suppose f is twice differentiable on Rand that there are three distinct points r1, x2, T3 in Rwith…

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Q: Prove that f (z) =zz is nowhere analytic

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Q: Consider the function f : R³ → R defined as V(x1, x2, x3) E R³ f(x1, x2, X3) = X1X2+ X1X3 + X2X3 a.…

A: We are given the function f: ℝ3→ℝ defined as ∀x1, x2, x3∈ℝ3    fx1, x2, x3=x1x2+x1x3+x2x3. For (a),…

Q: A function f: Z x Z is defined as f(m, n) = 3n - 4m.  Verify whether this function is injective and…

A: f: Z x Z is defined as f(m, n) = 3n - 4m So, f(m, n) = f(a, b) implies, 3n - 4m = 3b - 4a i.e. 3(n…

Q: If f is a function defined from R to R, is given by f(x) = 5x - 3 then f -(x) is given by %3D -- 20…

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Q: 4. Let f be defined for all real x, and suppose that \fx) – f(v)| < (x – y)? for all x, y e R. Prove…

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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…

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Q: Consider a function f:C C, defined by f(z) = 2z2 Then which of the following is true? Check all that…

A: Injective ( one to one): If x1 not equal to x2 then fx1 not equal to fx2. If x1 = x2 then fx1 = fx2.…

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

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Q: 5. Determine whether the closed interval [a,b]is homeomorphism to the closed unit…

A: This is a problem from topology.

Q: f(x)=xmodx for all x  in R. Discuss the derivability of f(x) at x=0

A: Given:fx=xxwe have to find derivability of fx at x=0

Q: Show that f : ℝ → ℝ, f(x) = 5x − 3 is a bijective function.

A: The given function is f:ℝ→ℝ defined by fx=5x-3. We say that a function f is bijective if it is both…

Q: Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c…

A: Intemediate value theorem gives that if f(x) is a continuous finction within the interval a, b  then…

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

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Q: Find aland U so that the function S 7z³ – 3x² + 7, z -2 is both continuous and differentiable. al =

A: To Determine: fins a and b so that the function is continuous and differentiable given: we have a…

Q: The function on ℤ defined by f(x)=-x^2-5 is onto. True or False ?

A: The given function is                  f(x) =x2-5   Graph:

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

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Q: If f is differentiable and non-negative on an interval I in R then /3f + 1 is differentiable on I.

A: The given statement is true. Because when f will be non negative then the derivative of the given…

Q: Define f: [0,2]-R by f(x) = x²+2x-1 , then the sup f- inf f is : [0,2] [0,2] .a 3 .b 8 .C 8, .d 1

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Q: Let f:A at f is surjective. B and g: B -C be two functions. If gof is surjective and g is total…

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

A: In mathematics, an injective function is a function f that maps distinct elements to distinct…

Q: Suppose that f is analytic on a domain D andVz € D,f'(z) = af(z) for some constant a. Show that f(2)…

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Q: There is a number c in the interval [a, b] such that f(c) is the absolute minimum of f on [a, b]

A: Given query is to find statement is true or false.

Q: Suppose that f(x) is convex on an interval I. Show that f(x1) + f(x2) + ·+ f(xn) n for arbitrary n…

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Q: Use FTC I to prove that if |f '(x)| ≤ K for x ∈ [a, b], then |f (x) − f (a)| ≤ K|x − a| for x ∈ [a,…

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Q: Let fi and f2 be concave functions from R" to R. Let f : R" +R be defined as the pointwise minimum…

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Q: Consider the function θ:{0,1}×N→Zθ:{0,1}×N→Z defined as θ(a,b)=a−2ab+bθ(a,b)=a−2ab+b. Is θθ…

A: Consider the given function as θ:0,1×N→Z defined as θa,b=a-2ab+b. Check for invective as follows: To…

Q: Prove that f(x) = x 3 - l2x is differentiable. Compute f'(x) and find f '(-3), f '(0), f' (2), and…

A: Given information: The function is f(x)=x3-x. Calculation: We need to find the value of f'(x) and…

Q: Verify that the Intermediate Value Theorem applies to the Indicated interval and find the value of c…

A: The given function is;  f(x)=x2+4x+2, [0,5], f(c)=23

Correlation

Correlation defines a relationship between two independent variables. It tells the degree to which variables move in relation to each other. When two sets of data are related to each other, there is a correlation between them.

Linear Correlation

A correlation is used to determine the relationships between numerical and categorical variables. In other words, it is an indicator of how things are connected to one another. The correlation analysis is the study of how variables are related. 

Regression Analysis

Regression analysis is a statistical method in which it estimates the relationship between a dependent variable and one or more independent variable. In simple terms dependent variable is called as outcome variable and independent variable is called as predictors. Regression analysis is one of the methods to find the trends in data. The independent variable used in Regression analysis is named Predictor variable. It offers data of an associated dependent variable regarding a particular outcome.