8. Show that the function f : N → Q defined by f(n) n- is one to one. n+1

Q: if and only if F-1: R(F) →X is bounded.

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Q: Suppose f(x) = 4nx2e Ax* for all x E R. Show that f = -f. %3D -e

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Q: 7. Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval:…

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Q: Prove that the function given by f(x) =x3-3x2+3x-100 is increasing in R

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Q: on satisfying f'(x) the function sat

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Q: 2. If p and q are positive integers, f is a function defined for positive numbers and attains only…

A: Introduction :

Q: Evaluate the function f(x) = e* + In (x + 7) at c, where –7 < c < 6.

A: Given information : f(x)=ex+ln(x+7) 'c' lies between -7 and 6

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 –…

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Q: Exercise 9.23. Is there an analytic function f R→ R for which f(n) (0) = n for all n ≥ 0? If there…

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Q: Prove that the function f(x) = x3-3x2+3x-100 is increasing on R

A: Given function is : f(x) = x3-3x2+3x-100 We have to prove that f(x) is increasing on R

Q: 2. Suppose that f is an increasing function on the interval [a, b]. Let F(t) = %3D a.) Argue that if…

A: suppose that f is an increasing function on the interva a,b let Ft=∫0tfxdx (a) if a&lt;x*&lt;b then…

Q: Let F be the collection of all functions f: N-→ {0,1}. Is F countable ?

A: A set S  is countable if there exists a one-to-one map from the set to the set of all-natural…

Q: Let T be a function from Z+ to Z defined recursively as follows: 1, if n is 1; T(n) := 2 + T(;), if…

A: Given T be a function from ℤ+ to ℤ defined recursively as follows, Tn=1,  if n is 12+Tn2,  if n is…

Q: 5. Consider the function f : N → N defined by 0 if the Riemann Hypothesis is true f(n) = · |1 if the…

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Q: (a) Assume that f(x) is a function defined by f(x) = x²-3x+1 2x - 1 for 2 ≤ x ≤ 3. Prove that f(x)…

A: Given.

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: 5. Suppose f(x) is an odd function. Prove that g(x)=f(x)| is an even function.

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Q: 5. Consider the function f : N → N defined by 0 if the Riemann Hypothesis is true f(n) = if the…

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Q: . Show that the function f: R→ R defined as f(x) = 13x − 15 is one-to-one.

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Q: 18.12. Let the function f be entire and f(z) → o as z 0o. Show that f must have at least one zero.

A: We are given that f be entire and f(z) goes to infinity as z →∞  and we  have show that f has at…

Q: Consider the following statements 1- The function f defined by f(x) = x³ is decreasing in R 2- If a…

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Q: For the function f (x) = |x| defined on [-1,1], there is some -1<c<1such that f' (c) = 0. O True…

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Q: (a) Let the function g : Z → Q be given by the rule 2n° +3 g(x) n2 + 1 Justifying your answers, say…

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Q: 1. Define function f : Z+ → Z such that f(n) = /n. (a) Is f one-to-one? Justify (prove) your answer.

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Q: Give an example of(a) a function f : Z → N that is both one-to-one and onto N; and(b) a function f :…

A: Here, we need to give an example of(a) a function f: Z → N that is both one-to-one and onto N.

Q: (b) Use the definition in (a) to compute f'(2) for the function f(x)

A:

Q: Suppose we have a function k(t) such that k(7) = 10 and k' (7) = –0.25. %3D

A: b) Assume k(t) = At2 + Bt + C

Q: Assume that f(x) is a function defined by x²-3x+1 f(x) = 2x - 1 for 2 ≤ x ≤ 3. Prove that f(x) is…

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: Show that the function f: N→N, defined by x + 1, if x is odd f(x) = x – 1, if x is even is one-one…

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Q: 10.18. Let A = {w, x, y, z} and B= {r, s,t}.Give an example of a function f: A →B that is neither…

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Q: If we apply Rolle's Theorem to the function f(x) = 2x- 16x 6 on the interval [2, 6|, how many values…

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Q: 4. Give an example of a function from N to N that is (a) one-to-one but not onto. (b) onto but not…

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Q: 1. Define h(x) = +y7x + 2, for all real numbers x > -. (a) Is h a one – to – one function? Why or…

A: Since not a particular question asked ,as per guidelines solution to only first question is given

Q: 4. Suppose that f and g are continuous on the closed interval [a,b] and dif- ferentiable on the open…

A: That's easy, you just need to use Lagranges mean value theorem and given conditions and then the…

Q: Suppose f(x) = 4nx?e-Tx -e-n for all x € R. Show that f = -f. %3D

A:

Q: For any positive integer n show that 1 n + 1 holds for any f e C'[0, 1] satisfying f(0) = 0. Verify…

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

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Q: Find a function f ( x) such that 5+ f(t) -dt = 4/x for and some number a а.

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Q: Define a function g(x) of a random 1; if x2x, 0; otherwise variables by g(x) = %3D Where x, is a…

A:

Q: Draw the graph of the function f(n)= 1– n from Z to Z.

A: Graph of the function

Q: 3. Give an example of a function f.N→N which is surjective but not injective.

A:

Q: 1. Suppose that f2(x) is a measurable function. Is it true that f(r) is measurable? Prove, or give a…

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Q: Suppose the function f has the property that there exists a number B such that |S(x) – f(c)| < B|x –…

A: Suppose the function f has the property that there exists a number B such that |f(x)-f(c)|≤B|x – c|…

Q: Show that a function f(x) = ∑ ∞n=2 (1/x-nπ + 1/nπ) is well defined on [0, π]. That is, prove the…

A: Given function,

Q: 6 Suppose that a function h satisfies –x² < h(x) < x² for all x. Prove that h is differen- tiable at…

A: Definition used: f'x=limh→0fx+h-fxh and f'a=limh→0fa+h-fah Calculation: Consider the given…

Q: Consider a function f defined from the interval [4, 7] to the interval [4, 7]. Which of the…

A:

Q: A function f :N Z which is one-to-one and onto.

A: Suppose we have a function . Condition of f is one-one . Suppose such that , then . Condition…

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: Let f′(t) &lt; 0 for all t in the interval (2, 8). Explain why f(3) &gt; f(5).

A:

Q: Find e so that f(x) is continuous on the entire number line. if x3

A: fx=x2    if  x≤3cx    if  x&gt;3 Since the given function is a polynomial function for all x&lt;3…

Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!