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c) Bijective (both injective and surjective) every x gives a unique value for 3x+2. so, this is injective(one-to-one) f(x)=y=3x+2 f^-1(y) = (y-2)/3 inverse of this function exits. so, this is surjective as well. so, this is both injective and surjective. hence this is bijective
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Let R represent the set of all real numbers. Suppose f:R -> R has the rule...
5. Let A = P(R). Define f : R → A by the formula f(x) =...
5. Let A = P(R). Define f : R → A by the formula f(x) = {y E RIy2 < x). (a) Find f(2). (b) Is f injective, surjective, both (bijective), or neither? Z given by f(u)n+l, ifn is even n - 3, if n is odd 6. Consider the function f : Z → Z given by f(n) = (a) Is f injective? Prove your answer. (b) Is f surjective? Prove your answer
Let the universal set be R, the set of all real numbers, and let A {xE...
Let the universal set be R, the set of all real numbers, and let A {xE R I -3 sxs 0, B {xER -1< x 2}, and C xE R | 5<xs 7}. Find each of the following: (a) AUB {xR-3 < x2} s -3orx > 과 xs. (b) AnB xR-12 {*E찌-1 <xs마 frER< -1 orx {*ER|x s -1 or*> 아 (c) A {*ER-3 <x< 아} {*ER|-3 < 아} s-3 orx> 아 frER< 3 orx x s 0 (d) AUC...
2. Let a be a positive real number, let r be a real number satisfying r...
2. Let a be a positive real number, let r be a real number satisfying r >1, let N be an integer greater than one, and let tR -R be the integrable simple function defined such that tr,N(r) = 0 whenver x < a or z > ar*, tr,N(a) = a-2 and tr,N(z) = (ar)-2 whenever arj-ıく < ar] for some integer j satisfying 1 < j < N. Determine the value of JR trN(x) dz.
12 if x = 1,2 1. Define f:[0,2] → R by the rule f(x) = {...
12 if x = 1,2 1. Define f:[0,2] → R by the rule f(x) = { 11 otherwise a. For any e > 0, find a partition Psuch that U (f, Pc) < € (be careful, as the minimum value for the function is one and not zero) b. Evaluate ſf
k=42, m=18 n=4 11. Let F:R → R be a function such that (t+m)(n+1) (n+ m...
k=42, m=18 n=4
11. Let F:R → R be a function such that (t+m)(n+1) (n+ m F(t) = for t <-m, f or-m <t<n. for n<t<k, for t > k. nA - 1 Find A and B knowing that F is the cumulative distribution function of a random variable X such that P(X = k) = . Please provide only the value of parameter B in the space specified below. ANSWER: B= Solution:
8. Let f:D → R and let c be an accumulation point of D. Suppose that...
8. Let f:D → R and let c be an accumulation point of D. Suppose that lim - cf(x) > 1. Prove that there exists a deleted neighborhood U of c such that f(x) > 1 for all 3 € Un D.
7. Consider the function f:R + R defined by f(x) = x < 0, 3 >...
7. Consider the function f:R + R defined by f(x) = x < 0, 3 > 0. e-1/x2, Prove that f is differentiable of all orders and that f(n)(0) = 0 for all n e N. Conclude that f does not have a convergent power series expansion En Anx" for x near the origin. [We will see later in this class that this is impossible for holomorphic functions, namely being (complex) differentiable implies that there is always a convergent power...
Let f and g be differentiable on R such that f(1) = g(1), and f'(x) <...
Let f and g be differentiable on R such that f(1) = g(1), and f'(x) < '() for all r ER. Prove that f(x) = g(2) for 3 >1.
Let z=5 where x, y, z E R. Prove that z? +z2+z?>
Let z=5 where x, y, z E R. Prove that z? +z2+z?>
Let U be an open subset of R". Let f: UCR" ->Rm. (a) Prove that f...
Let U be an open subset of R". Let f: UCR" ->Rm. (a) Prove that f is continuously differentiable if and only if for each a e U, for eache > 0, there exists o > 0 such that for each xe U, if ||x - a| << ô, then |Df (x) Df(a)| < e.
5. Let A = P(R). Define f : R → A by the formula f(x) = {y E RIy2 < x). (a) Find f(2). (b) Is f injective, surjective, both (bijective), or neither? Z given by f(u)n+l, ifn is even n - 3, if n is odd 6. Consider the function f : Z → Z given by f(n) = (a) Is f injective? Prove your answer. (b) Is f surjective? Prove your answer
Let the universal set be R, the set of all real numbers, and let A {xE R I -3 sxs 0, B {xER -1< x 2}, and C xE R | 5<xs 7}. Find each of the following: (a) AUB {xR-3 < x2} s -3orx > 과 xs. (b) AnB xR-12 {*E찌-1 <xs마 frER< -1 orx {*ER|x s -1 or*> 아 (c) A {*ER-3 <x< 아} {*ER|-3 < 아} s-3 orx> 아 frER< 3 orx x s 0 (d) AUC...
2. Let a be a positive real number, let r be a real number satisfying r >1, let N be an integer greater than one, and let tR -R be the integrable simple function defined such that tr,N(r) = 0 whenver x < a or z > ar*, tr,N(a) = a-2 and tr,N(z) = (ar)-2 whenever arj-ıく < ar] for some integer j satisfying 1 < j < N. Determine the value of JR trN(x) dz.
12 if x = 1,2 1. Define f:[0,2] → R by the rule f(x) = { 11 otherwise a. For any e > 0, find a partition Psuch that U (f, Pc) < € (be careful, as the minimum value for the function is one and not zero) b. Evaluate ſf
k=42, m=18 n=4
11. Let F:R → R be a function such that (t+m)(n+1) (n+ m F(t) = for t <-m, f or-m <t<n. for n<t<k, for t > k. nA - 1 Find A and B knowing that F is the cumulative distribution function of a random variable X such that P(X = k) = . Please provide only the value of parameter B in the space specified below. ANSWER: B= Solution:
8. Let f:D → R and let c be an accumulation point of D. Suppose that lim - cf(x) > 1. Prove that there exists a deleted neighborhood U of c such that f(x) > 1 for all 3 € Un D.
7. Consider the function f:R + R defined by f(x) = x < 0, 3 > 0. e-1/x2, Prove that f is differentiable of all orders and that f(n)(0) = 0 for all n e N. Conclude that f does not have a convergent power series expansion En Anx" for x near the origin. [We will see later in this class that this is impossible for holomorphic functions, namely being (complex) differentiable implies that there is always a convergent power...
Let f and g be differentiable on R such that f(1) = g(1), and f'(x) < '() for all r ER. Prove that f(x) = g(2) for 3 >1.
Let z=5 where x, y, z E R. Prove that z? +z2+z?>
Let U be an open subset of R". Let f: UCR" ->Rm. (a) Prove that f is continuously differentiable if and only if for each a e U, for eache > 0, there exists o > 0 such that for each xe U, if ||x - a| << ô, then |Df (x) Df(a)| < e.
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