8. Decide if the following functions are onto. Prove your answers. (a) f(x) = x² – 1 (b) g(x) = 2x +1

Q: True/False: If ƒ is entire and |f(z)| > 2 for all z E C with |2| > 1, then f is a constant function.

A: If f is entire and fz≥2 for all z∈ℂ with z≥1Let gz=1fzas fz≠0 for any z∈ℂ⇒g is entire gz=1fz<12…

Q: 2. Consider the function / with domain (-2, o0): 3|2 – 1| 2 + 27 - 4 if – 2 <z<1 f(z) = if r = 1…

A:

Q: If fx) is a continuous function on [4,7] and F(x) is an anti-derivative of f(x) such that F(4)=9 and…

A: Given f(x) is continuous on [4, 7] and F(x) is the anti-derivative of f(x)                 F (4) = 9…

Q: Define h(2) in a way that extends h(t) = (t2 + 3t - 10)/(t - 2) to be continuous at t = 2.

A: Consider the given the function,

Q: If L(x)=mx+b is the linearization of f(x)=2x³ - x +3 at x-1, then m= Using L to approximate 2(1.01)…

A:

Q: 6. Consider the function f Cx)=5(o+5)-25 if X#し if xfO D.IS fCx) continuous at X-0? EXPlain

A:

Q: 1 Q9: Let f(x) = cosx x + 3 then find all critical points. 2 -

A: A critical number C is a number in the domain of 'f' for which eitherf'c=0 or f'cdoes not exists.…

Q: 1- Discuss the continuity of the following functions at given poin { V9r4+x² 5x2+3x+1 (a) f(x) = if…

A: It is given that

Q: of of and Find for the following function. f(x.y) = e xy In (2y) of dy

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Q: Find the Wronskian for the set of functions.{e3x, sin 2x}

A:

Q: If A(x) = f(t)dt is increasing on [0,10], then f(x) must be increasing on [O,10]. True False

A: Given

Q: 9. 6) Suppose that g is continuous and that g(x) dx = 2 and g(x) dx = 16. Find g(x) dx.

A:

Q: 10. Consider the following function. 1 x } x > Find the values of a and b that make f(x)…

A:

Q: Determine the types of singularities of the function f(z)=1-e^2z/z^4

A:

Q: 15. If g(x) and h(x) are both antiderivatives of some func- tion f(x), then what can you say about…

A: Consider that gx and hx are both antiderivative of function fx. Find gx-hx.   Since gx and hx are…

Q: The function g(x) = ln(x³ – 1) defined on V2, 0) is not zero unless x 2.

A: Given statement

Q: 1 Let f(x) = 8 + In (t) dt, for x2 1. Find f'(2).

A:

Q: For all but two choices of a, the function (x) = x if xS a and x' ifx > a will be discontinuous at…

A:

Q: If W) is a continuous function on [4,7] and F(x) is an anti-derivative of f(x) such that F(4)=9 and…

A:

Q: 2. Supposethat f(x)=?βx? −3 if βx + 2 ifx≤2 x &gt; 2Find the value of β so that f(x) is continuous…

A:

Q: (2) The real part of Syy (@) and the real part of Syx (@) are even functions of o, i.e., Re[Sy (@)]…

A:

Q: 11) Consider the function h(x)=Vx+6. Decompose into two functions g(x) and f(x) where h(x)= f(g(x))…

A: Given, hx = x+6  We have decompose into two functions fx  and gx  where hx = fgx  and gx≠ x   and…

Q: 35. Given that f(z) is a continuous even function, then A 0 B D

A:

Q: Show (in terms of e - 6) that a function f : [2,7] R be defined by f(z) = V² +1 is uniformly…

A: In this question, we need to show the given function f(x) is uniformly continuous.…

Q: Sßx² – 3 (Вх + 2 if (b) Suppose that f(x) if Find the value of ß so that f(x) is continuous…

A:

Q: 3 B Consider the following function, z= x -2xyty't yn Lind the alues of ag Ddz,z,when redgyel dy 6.

A: Given : z = x2 -2xy + y3 + yw

Q: 14. Consider the function f (x) = x²e+x . f(x) has two inflection points at x = _ and x =_.

A:

Q: Solve the following sub-problems. (1) Define a function f : Z→ Z × Z by f(x) = (x*,x²). Is f a…

A: One One function A function f:A→B is said to be one one function if  for any x,y∈A such that x≠y…

Q: 9. Consider the function f: R → R, defined by (x²sin(1/x) ifx # 0 f(x) = if x = 0 Then a. f is not…

A:

Q: 4. At what value(s) of x is the function x + 2 x2 for x 1 f(x) = 3- x discontinuous?

A:

Q: Show that the function T(#) = E(,n+) is well-defined Problem 7. n=2 on [0, 7]. That is, prove that…

A: Domain: The domain of a function is set of all values for which is well defined. Given:

Q: 2.4 Use the definition of the derivative of complex function to show that f(z) is not dif-…

A: The Complex Derivative Definition: Let f be a function defined on a neighborhood of z0 ∈ ℂ. If…

Q: 3. Define f(2) in a way that extends f(s) = -8 to be continuous at s = 2. s2 – 4

A:

Q: 1. Where are the following functions continuous? x F(x) = ex + 4-x3 G(x) = 3x5 - 4x? H(x) = x-5x3…

A:

Q: 2. Suppose that f is a measurable function satisfying m({r €R: [f(x)| 2n}) < ne N. n3' Is f…

A: Given that f is a measurable function, with the following condition satisfied. m(x∈ℝ:…

Q: If g(g(x)) is continous at 0, then g(x) is continous at 0.  True or False?

A:

Q: 7.14. Show that if f = u+iv is analytic in a region S and u is a constant function (i.e.,…

A: Analytic function which is a constant

Q: 7. The function (x, y)= (2x* + y* )exp + xy' is homogenous in x and y of degree

A: .Given,f(x,y) = (2x4+y4) ey2x +x y3Put y= xSo we get,f(x,x)= (2x4+x4) ex2x +x x3 f(x,x)= (3x4) e0.5…

Q: Use Part 1 of the Fundamental Theorem of Calculus to find the g'(x) if: g(x) = 1 +t² dt 3x2

A:

Q: Compute the linearisation l(x) of f(x) = √1+x3 at x=2 and use this to estimate f(2.05)

A: Given function is: fx=1+x3 At x0=2 So the linear approximation is: Lx≈fx0+f'x0x-x0…

Q: 2. Evaluate f(x) = e3x + 2 at x = -1. Round to the nearest thousandth. %3D %3D f(-1) = ...

A:

Q: 1. Consider the function z = f(x, y). If fa(a, b) and fy(a, b) both exist, and they are equal, then…

A: 1. Given: z=fx,y and suppose that fxa,b and fya,b both exists and they are equal and it is…

Q: 5. Consider the function S 2r – 1, si x < 2 f(x) = 6-x, si x 2 2. The Limit x 2 f(x) is:

A:

Q: Q.4 A/ Prove the function continuous at x = -2, (x2 + 5 13-3x where %3D if x<-2 if x 2-2 f(x) = %3D

A:

Q: If f(x) = then f (x) at z = 1 is

A: Given that :-         f(x)=1x         f'(x)=? at x=1

Q: 2. Determine whether the following functions are linearly dependent or not by applying the…

A:

Q: ( al=) if x # 0 1ifx 0 6. Let f(x) = borln Find f'(0).

A:

Q: Suppose f(x) is a bounded real - valued function on [a.b) such that fela,b] . Does it follows that…

A: A function is bounded in [a,b] if it can't achieve value beyond a and b. i.e.maximum acheivable…

Q: 1. If y = f(x) is a continuous function for all 0 <x< 6, and f(0) = 1 and f(6) = 7, then which of…

A: We use the given graph of the function y=f(x), to answer the given question.