(3) Suppose that f: [0, 1] → R is a real function. with f([0, 1]) ≤ [0, 1]. Show that there exists an a € [0, 1] with f(a) = a. (Hint: Consider the function g(x) = x − f(x).)

Q: Let f : R R:f(x) = 10x + 7. Find the function g : R→R such that gof = fog = IR. %3D %3D

A: Given f: R -> R  such that f(x) = 10x + 7 and  g : R -> R such that g o f = f o g = IR

Q: x(x²+3) Q7. Consider the function f : [0, 1] → R, → √√4+24 Explain why (a) f continuous, (b) f is…

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Q: 4. If f:(X,r)(Y.A) and g:(Y.A) (Z,r)are closed functions, prove that g of :(X.r)(Z.r) is closed…

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Q: Give an explicit increasing linear bijection from the interval (a, b] (a < b) to (0, 1]. i.e. your…

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Q: 3. Find the fixed point(s) of f : [1, 0) → [2, 0) defined by f(x) = vx² + x.

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Q: The function f from R to R is defined as f (x) = x²(x − 1), determine (a) whether f is one to one.…

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Q: 5. Define f: R→→ R and g: R → R defined by f(x) = max{-x, x} and g(x) = |x|, respectively. Prove…

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Q: Q4E. f:(0, 19/4] R defined by f(x)=(19–4x). Prove that f is a one to one function mapping onto [0,…

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Q: Give an example where | f(x)dg(x) exist even through f is unbounded on [a, b].

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Q: Is there an unbounded function f:[0,1] →R where it is discontinuous only at a single point in [0,1…

A: f: [0, 1] →R Domain: [0, 1] The function is discontinuous at a single point of the domain.   Let's…

Q: Let f: R (-1, 1) be a function defined by Prove that (a) f is injective; (b) f is surjective. f(x) =…

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Q: 1. Define f : (- o, 2) → (- 0, 1) by f(x) = . Show that f is bijective. x+1 *-2

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Q: Suppose a smooth function f(x) satisfies f(0) = 0, f(1) = 2, and that f[x0,x1,x2] = 1  for any three…

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Q: Let ƒ : R\{1} →→ R\{1} be defined by f(x) = x/(x - 1). Show that ƒ is bijective and find f-¹. Also,…

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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: Let f: Z → Z be defined by f(x) = x² + 1, and let C = {1,2,3}.

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Q: 3. Let X = {1,2, 3, 4}, Y = {a, b, c} and let f: XY. Prove by contradic- tion that f can't have an…

A: Given that: X={1,2,3,4}and Y={a, b, c}and f:X→Y

Q: 1. Define function f : Z+ → Z such that f(n) = /n. (a) Is f one-to-one? Justify (prove) your answer.

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Q: Q.10. Let [a, b] be an interval of R. Let f:[a,b]→R be a function such that f is M-times…

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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: (2) Let f(x) = ax³ + bx? + cr + d, where a, b, c, d e R and a + 0. Prove that f(x) must have a root.…

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Q: Consider the function ƒ : R → R defined by 2x + 5, ax² + x, Show that there is a value of a for…

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Q: a)Suppose f: R → Z, where f(x) = [x + ¹] i. Graph the function f(x). ii. If A = {x: 1 ≤ x ≤ 3}, find…

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Q: 6. Let f(x) = x55x+1. Prove that f has at least 3 zeros in [-2, 2]; that is, there exist distinct…

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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: Let f be an analytic function on C: |z-1+i=2 with |f] 4 b. [f"(1 – i)| > 2 c. If"(1 – i)| < 2 d.…

A: here option (b) is true.

Q: Let f be an analytic function on C: z-1+i=2 with If] 4 b. |f"(1 – i)| > 2 c. If"(1 – i)| < 2 d. None…

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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…