Let function f: Z Z be defined as follows; 1 x is odd f(x) = . x² + 2 x is even The functionf is . a) 1-1 but not onto b) onto but not 1-1 c) 1-1 and onto d) neither 1-1 nor onto

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A: There are many types of relations Symmetric Asymmetric Reflexive Transitive

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A: Please see the attachment

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A: A set F is said to be a compact set if it is closed and bounded both.

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A: f is not one-one  But, f is onto.

Q: Prove that f(x) = 5x2 + 3x – 26 is θ(x2)

A: Given: f(x) = 5x2 + 3x – 26 is θ(x2)

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A: First option is correct

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Q: Give an example of onto but not 1-1 function f: 7⁰ → Z>⁰. tit 11-C.

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Q: If X and Y are finite sets, prove that XY = |X|Y. Here, XY is the set of all functions from Y to X.

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Q: Let z be the set of integers and f:Z → Z be defined as f(x)=-4x+5. Then the inverse function of f *…

A: As per our company's policy, we can give answer to only first question. If you want answer of the…

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Q: Suppose f and g are continuous on a closed interval [a, b] such that f(a) > g(a) and f(b) < g(b).…

A: I have used intermidiate value theoram of continuous function.

Q: (a) Prove that there exists a function f : Z → Z so that f is one-to-one but f is not onto. (b)…

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Q: Let f : [a, b] → R be a continuous function and suppose that N is a natural number. If ¤1,..., IN…

A: It's a question from intermediate value theorem, so let's see how we can do it.

Q: Let A be a finite set, and suppose that f : A → A is one-to-one. Does f necessarily have to be onto?…

A: The given question is solved below.

Q: Prove that for functions f : R→R, the e- 6 definition of continuity implies the open set definition.

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Q: Define f:Z* x Z* →z* by the formula f(m, n)= 4"6" for all (m, n)Ɛ Z* × Z*. (a). Is f one-to-one?…

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Q: Let Z be the set of integers and f:Z -- Z be defined as f(x)=-2x-3. Then the inverse function of f *…

A: Let  f:ℤ→ℤ  given as fx=-2x-3.

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Q: Show that the function f: Real #s to Real #s defined by f(x)=x2 is not a homomorphism.

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Q: A function g: Z → Z is defined by the rule g(m,n)=m+n Determine whether it is onto or not...

A: Onto functions: A function f: A→B is said to be an onto function if range of function is equal to…

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Q: 4. Define f:R → R, define f (x) = 3x + 5 for all x e R. Show that a. fis one - to - one; b. f maps R…

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Q: Let f be a differentiable real function defined in (a, b). Prove that f is convex if and only if f'…

A: A function f is convex in a domain a,b if for all 0≤λ≤1, fλx1+(1-λ)x2≤λfx1+1-λfx2  ∀x1,x2∈a,b. Also…

Q: Prove or disprove the relation C on Z defined by C = {(x, y): x=y (mod 5)} satisfies the two…

A: I just used the definition of congruence and divisibility

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Q: 2. (a) Assume that (a, b) is a bounded open interval in R. Using Mean Value Theorem, prove that if…

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

Q: 3. Consider the function f: IR→ (0,1] defined as f(x) = x² +1 (a) Prove that f is not one-to-one.…

A: Given function is f: ℝ → (0, 1] such that f(x) = 1x2+1

Q: 2. Let f: R → R_be f(x) = 1+ |r|" (a) Is f everywhere defined? If not, give the domain. (b) Is f…

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Q: | Given a finite set A with n elements, let R be the relation defined on the set of functions f:A →…

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Q: (a) Prove that there exists a function f: Z → Z so that f is one-to-one but f is not onto. (b) Prove…

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Q: Let f : R + R defined by f(x1, x2)) = x1 - 2. Then determine if is (a) Lipschitz continuous(if it…

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Q: Exercise 11: Suppose that f and g are continuous on a, b, and f(a) g(b). Prove that there exists c…

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A: Given  ∂f∂x=xy   ⋯⋯1∂f∂y=x2   ⋯⋯2.

Q: Let z be the set of integers and f:Z → Z be defined as f(x)=-4x+5. Then the inverse function of f *…

A: We know that inverse of a function exist if it is has one to one correspondence and it is onto. If…

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Q: Determine whether the function f: Z x Z -→ Z is onto if f (m, n) = .m + n

A: We will solve the problem using the definition of onto function.

Q: Define the function f: R→ R by f(x) = x5 – 1, Vx € R. a) Prove that f is onto? b) Prove that f is…

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Q: Let f: [a, b] → [a, b] be continuous. A fixed point of f is a member x E [a, b] such that x = f(x).…

A: Here we use IVP theorem to prove this

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Q: (a) Prove that there exists a function f : Z→ Z so that f is one-to-one but f is not onto. (b) Prove…

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A: Given, The function f is continuous in the interval 0,1, X=x∈0,1: f is strictly increasing in 0,x…

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: