Consider the following function: f: Z → N given by f(x) = (x+4)² + 8; where, as usual, Z and N, denote the set of all integers and the set of all natural numbers, respectively. You are given that the function f is not an injection. In order to show that f is not an injection, state two distinct elements of the domain of f, denoted by x1, x2, such that f(x₁) = f(x₂). • State 1: State ₂: You are given that the function f is not a surjection. In order to show that f is not a surjection, state an element of the codomain of f, denoted by y that is not in the image of f. • State y: I

Q: Find an example of a function ƒ: N → Z which is (a) injective but not surjective (b) neither…

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Q: Consider the functions f : A → B, g : B → A and h : B → A such that if gf = 1A and fh = 1B. Prove…

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Q: Consider the functions f : A → B, g : B → A and h : B → A such that if gf = 1 Ą and fh = 1B. Prove…

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Q: You can get a car loan with a term of three years at an apr of 3%. If you can afford a monthly…

A: Given: interest rate per annum is= 3%, monthly payment= $300, time=3 years To find the amount of…

Q: 4. Lėt f:A total injective function. B and g: B→C be two functions. If gof is a bijective function,…

A: we know that  gof is Bijective,  hence, gof has one to one correspondence,  This deduces that g must…

Q: b) Let the set A = {-1,0, 1} and consider the following function: f2 A → A given by f₂(x) = (-x)4.…

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Q: A function f : Z Z is defined by f(n) = 3n + 1. Let E be the set of even %3D integers. Determine the…

A: When a function is linear, it is one-one and onto function. We know that a bijective function is…

Q: 2. Let X1 and X2 be sets. Define functions T1 : X1 × X2 → X1, T2 : X1 × X2 → X2 by T| (X1, X2) = x1,…

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Q: Let f(x)= Ax² + Bx+C (A, B, C e R). Prove that if f(x) is an integer whenever x is an integer, then…

A: Given :   f(x) = Ax2+Bx+C,   where  A, B, C ∈ ℝ. To show :  If f(x) is an integer whenever x is an…

Q: Consider the recursively defined function: f(0) = 2 , f(n) = 2f(n-1) +5 where n   is an integer…

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Q: Consider the set of all functions f : N → {0, 1}, i.e., the functions map the natural numbers to 0…

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Q: 2. Let R be the relation on the set A = (1, 2, 3, 4, 5, 6, 7} defined by the rule (a,b) e R if the…

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Q: d) Let the set A = {1,2,3,4} and let P(A) denote the power-set of A. Consider the following…

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Q: Let f:R → R be the function with rule f(x) = 5x + 2, and let g: R → R be the function with rule y –…

A: To find  f0g(-1)  And  f0g(6)

Q: Let R = {(n, −n) | n ∈ Z and n ≤ 0} ∪ {(n, n + 1) | n ∈ N}. Is R a function? Justify your answer.

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Q: Let g : B → C and h : B → C be functions. Suppose that go f = hof for every function f: A → B. Prove…

A: We have to prove that g(x) = h(x) for all x belongs to B.

Q: Determine whether each of these functions from R to R is a bijection.  f(x)=2x+4

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Q: Let f : A →B and g : B →C denote functions, and assume g ◦f is injective. a. Prove or disprove: The…

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Q: For any subset A of R, define the function XA : R →R by if x E A 1, XA(") = 0, if x ¢ A. Let A and B…

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Q: Let A = {1,2,...,n} and consider the functions f: A → A. a. How many functions satisfy f(1) = 1? b.…

A: This question related to function and bijection.

Q: 2. The functionsf and g are defined on R by f:x + 2x-1 X-2 A B g:x 2 C g:xH x+1 D g:x+

A: Under the given conditions we need to find the function g :

Q: (a) Give an example of two functions f: A→B and g: B-C such that f is both 1-1 and onto but gof is…

A: (a)let's define sets 𝐴, 𝐵, and 𝐶 as follows:Let 𝐴={1,2}Let 𝐵={3,4}Let 𝐶={5}Now, let's define…

Q: Express the following functions F, H and G in the form f∘g. (Enter your answers as a comma-separated…

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Q: Let f: R+ → N+ with f(x) = [x²]; that is, f(x) returns the square of x rounded up. Characterize f in…

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Q: CONSIDER A FUNCTION OF R IN R DEFINED BY: x² + 1 X-2 SO, THE EXPRESSION: f(x) - f(3) x - 3 FOR 2 IS…

A: f is a function of R in R defined by  f(x)=( x2+1)/(x-2)  Then expression [ f(x)-f(3)]/(x-3) ?

Q: Let A = {a, b, c, d, e}, B = {1, 2, 3, 4, 5} and let us consider the function f : A → B, f(a) = 3,…

A: Given that:  f : A → B, f(a) = 3, f(b) = 4, f(c) = 2, f(d) = 5, f(e) = 1.   Since, the images (i.e.…

Q: Let f be a function, and suppose that A is a subset of the domain of f. The image of A under f is…

A: Let f be a function, and suppose that A is a subset of the domain of f. The image of A under f is…

Q: Let A = Z and B = Z. Suppose f: A ∈ B is defined by the rule f(a) = a3 + a + 1. Determine f(−1) and…

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Q: Let f : A → B and g : B C be surjective functions. Prove or disprove whether gof is surjective.

A: Here given that                       f :A→B   and    g :B→C  be both surjective function .  Now we…

Q: b) 0 1 2 3 4 5 y 8 3 0 -1 0 3

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Q: a) Consider the following function: f₁ ZZ given by f₁(x) = -3x7 +5. Decide whether this function is…

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Q: The functions h and m are defined on the set R of real numbers by h:x→ ax +b and m:x → cx -d, where…

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Q: Let R be the set of all real numbers. Determine every function g: R→ R such that for all real…

A: We need to find all functions g : ℝ→ℝ that satisfies g(a2+ b+g(b))=2b+(g(a))2.

Q: Construct the function f(r) = r+2 on the set of points x = 0 to 2 in steps of 0.1 and give the value…

A: We have function  f(x) = x2+2  Now ,  Sr. No. Value of  x Value of function f(x) =y 1. 0 f(0) =…

Q: D. Let function g: R → R be defined by g(x) = 2x + 1 5 cx in its domain. Why does your choice of c…

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