5. Let A = {a,b} and B = {1,2,3}. (a) Write out all functions f: A → B using two-line notation. How many different functions are there, and why does this number make sense? (You might want to consider the multiplicative principle here). (b) How many of the functions are injective? How many are surjective? Identify these (circle/square the functions in your list).

Q: Find a closed form representation for the function defined recursively by ƒ(1)=4 and f(n)=3f(x)+n.

A: This is a question of recurrence relation and time of complexity.

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A: Multiplicative function: A function is said to be multiplicative if , where . Given: The function…

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A: Find your answer below

Q: Find a closed form representation for the function defined recursively by f(1) = 10 f(n)=2ƒ(%)+n².

A:

Q: 7. If a function f, whose domain is all real numbers, is even and if 4 is a zero of f, then is also…

A: Given : if a function f , whose domain is all real numbers is even and if 4 is a zero of f then…

Q: C. Show directly from the definition (without using Arithmetic of Limits or similar results) that…

A: Topic:- continuous function

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A: Given :   Fx = x2+1              x<12                     x=18-3x              x>1 To find :…

Q: Construct the table of signs for f'(x) and f"(x) ginve f(x)= x4ex on its domain of (-inf, +inf)

A:

Q: e, F. g}. Define functions H and K by the arrow diagrams below. Domain of H Co-domain of H Y H od of…

A:

Q: 6. (a) Use the Schröder-Bernstein Theorem to prove that [a, b]| = |(a,b)| for any a, b e R. (b) Use…

A:

Q: 5. Check the continuity of r+5| 2x+10 x + -5 f(x) = x = -5 at x = -5.

A:

Q: (a) Someone has claimed that since 14 is itself a multiple of 7, then there are more multiples of 7…

A:

Q: 1 Expand f(2) = -in the domain 4 < |2 < 12 (z-4)(z–12)

A:

Q: Suppose f is a function with domain [a, b]. If f is an odd function, what can be deduced about a and…

A: Since solve the first question for you. If you want any specific question to be solved then please…

Q: 5. 6. Let f(r) = a. Show that f is one-to-one. That is, show that if = , then a = b. %3D b. Compute…

A:

Q: - whether the following relation is a functio- x² – 2x + 1

A: We have to identify  the given  relation is function  not

Q: 16.1. Give an example of a bounded and continuous function g on R to R which does not take on either…

A: Note: Since you have asked multiple questions, we will solve the first question for you. If you want…

Q: 2. For each of the following, give an example or explain briefly why no example is possible. (a) The…

A: Given

Q: 12. Consider the functic f(x) = x°, g(x) = %3D a. Give the domain b. For each function inverse. c.…

A:

Q: A relation R exists such that for every input(x) is 2x. Is relation R a function, and why (20…

A: Solution - Option C is correct, "The relation R is a function because there is only one output for…

Q: 1 Expand f (2) = in the domain 2 < |z| < 14 (z-2)(z-14)

A: Follow the procedure given below.

Q: 1. Find all possible points of discontinuity and classify each discontinuity as removable, jump…

A:

Q: If f A —> B and g : B —> C are injective function, then gof: A —> C is an injective function. Prove…

A: Given that f: A→B is given injective. this implies cardinality of set A is less then cardinality of…

Q: 6. Determine the algebraic representation of f-(x) from the tabular representation of ƒ(x) and…

A: If  x,y is a coordinate of the function f(x), then y,x is the coordinates of f-1(x). The coordinates…

Q: 17. Find the inverse of f(x) (x+2)2-6 for x 2 -2. Include a domain restriction if appropriate.

A:

Q: (1) Suppose that f and g are multiplicative functions. Prove that the function F defined by I…

A: Consider functions f and g to be multiplicative functions. To prove that function F defined by…

Q: Find the domain of the function. k(x) = In |x|

A: The domain of a function, f(x)  is the set of input values of x for which f(x) is real and defined.

Q: If f(x) =x²-x over x-1 and g(x) =x is it true that f=g?

A:

Q: How do you show the image of the union of S and T is equal to the union of the image of S and the…

A:

Q: 2. Determine whether Rolle's Theorem can be applied tof on the closed interval [a,b]. If Rolle's…

A: Here rolles theorem is applicable.

Q: Sketch a possible function that satisfies the following conditoons: a) Quartic, as x goes to…

A: It is known that a polynomial function with n degrees has at most n zeros and n-1 turning points.…

Q: Suppose, we want to define functions from set Ato set B Given that, |B| = n= 20 > 15 = m =HAN many…

A:

Q: 5. The set of all pairs of real numbers of the form (x, y), where x ≥ 0, with the standard…

A:

Q: 2. If f : A → B and g : B→ C are injective functions, then gof: A → C is also injective.

A: We need to prove/disprove statenemt two.

Q: 6. Based on what you know about the multiplicative principle, how many functions f:A- Bare there if…

A: Solution:

Q: 1. Find all possible points of discontinuity and classify each discontinuity as removable, jump…

A: Let's find.

Q: 8. 7 If f(x) = and g(x) = - , %3D x²-x-2 find (f- g)(x) and determine any domain restrictions x2+x-6

A:

Q: 2. (a) Suppose f is a function with domain a, b. If f is an odd function, what can be deduced about…

A:

Q: Given any set of 15 integers, must there be two that have the same remainder when divided by 12?…

A: We have 12 remainders when a integer is divided by 12 those are 0,1,2,3,4,5,6,7,8,9,10,11   Let us…

Q: If B is countable and f : P(A) → B is a one to one (i.e., injective) function, then A must be…

A: We have to find is the statement is true or false.

Q: 5. Consider the function Ax) = - 2x + 3x - 8. Determine * + 4).[ 3A]

A: 5. Given function   f(x)=-2x2+3x-8 We need to find f(k+4)

Q: 1. Let A = {a,b,c} and B = {1,2, 3}.a) Write out all functions f : A → B using two-line notation.…

A:

Q: Let f and g be the function from the set of integers to itself, defined by f(x) = 2x + 1 and g(x) =…

A: Consider the two given functions. fx=2x+1 gx=3x+4   The composition of the functions f and g  is…

Q: Suppose that the functionh is defined, for all real number 1 2 x++5 if x -2 h (x) = < %3D -3 if x…

A:

Q: 8. Determine the algebraic representation of the following piecewise function. 4 3

A: Solution: Now it is known that the equation of parabola is given by y=ax-h2+k Where, h is the…

Q: 1. Find all possible points of discontinuity and classify each discontinuity as removable, jump…

A: This is a limit and discontinuity problem

Q: 8. Find a closed form representation for the function defined recursively by f(1) = 10 and…

A: Definition: Here is the master's theorem for recurrence relation is follows, Tn=aTnb+cnk, T1=c1…

Q: 11. The function f: R – {-2} → R is defined by f(x) : x+2 Determine f'(1) and verify your answer…

A: First we have to find  f'(1) and have to verify it by € _ delta definition.

Q: 3. Prove that the function f: R – {3} → R - {1} defined by f(x) = , is one-to-one correspondence.

A:

Q: The following functions all have domain {1,2,3,4,5}{1,2,3,4,5} and codomain {1,2,3}.{1,2 For each,…

A: