Solve the following sub-problems. (1) Define a function f : Z→ Z × Z by f(x) = (x*,x²). Is f a one-to-one function? Prove or disprove. (2) Define a function f : Z → Z × Z by f(x) = (x – 2,x²). Does ƒ map Z onto Z × Z? Prove or disprove.

Q: Let f : R → R be a function with the following property: |f (a) – f(b)| < 4|a – b| for all a, b E R.

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Q: Consider the functions from R to R defined by f(x) = x² + 3x - 2 and g(x) = 2x+4 a. What is fog(x)?…

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Q: Let f be the function defined above, where k is a constant. f(x) = x²+kx-2 3x²+4x+3 x³ + 2 0 2e 2-e…

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Q: (b) Define a function F:F: R XR → R XR as follows: For all (x, y) ERXR, (F(x, y) = (2x + y, 2x - y)…

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Q: 5. Prove that the function f: Z → Z defined by f(x) = 3x + 1 is one-to-one.

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Q: Let f : R → R be a function defined by f(x) = Decide whether it is an injection, a surjection, 1+ x2…

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Q: Show that the following relations are not functions on R. a) ƒ = {(x, y) = RxR | y=x/(x² −7x+12)}

A: Given function. y=xx2-7x+12 is not define for x2-7x+12=0 =&gt; (x-3)(x-4)=0

Q: (a) Let f:Q → Q be the function defined as follows: f(x) = 1 + 2x. Is f(x) a surjective function?…

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Q: Show that R(g ◦ f) ⊆ R(g). (b)  Give an example of two functions where R(g ◦ f) ⊂ R(g). Briefly…

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Q: Define f : R → R and g : R → R by the rules f(x) = 3.x – 2 for all x E R and g(x)= x² +3 for all x E…

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Q: Let f: R2 R2 be the function given by f(x, y) = (x + y, y). Prove or →>> 8. disprove: f is…

A: Function which is one to one

Q: Consider the function f: R → R defined by f(x) = x³ – 3x + 2. Then (A) ƒ is increasing on the…

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Q: 4. (a) (b) Let f : X → Y and g: Y→ Z be functions. Prove the following statements. If gof is…

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Q: Prove or disprove: f : Z30 → Z15 given by f([x]) = [x²] is a well-defined function.

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Q: b) Prove that if f(z) and f(Z) are analytic in a domain I, then I is a constant function.

A: Given Data: We have to prove  if fz and fz¯ are analytic in a domain Ω, then f is a constant…

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.