function g : Z → Z is defined as g(n) = |n| + 1. Show that this function is not injective and not surjective.

Q: A function f is choice of x1 and x2 in I, with xi<x2, then f(x) <f(x2). on an interval I if, for any

A: we have to fill the given blank statement. Statement : A function f is ____________ on an interval I…

Q: Sketch the graph of a continuous function y = g(x) such that a. g(2) = 2,0 2, and g'(x) →-1* as…

A: It is given that

Q: (a) Consider a function f: R → R whose graph Gf is a closed subset of R?. Use the sequential…

A: For (a), We are given a function f : ℝ→ℝ whose graph Gf is a closed subset of ℝ2.  If f is also…

Q: Let f:R-->R and suppose that |f(x)-f(y)| less than or equal to |x-y|^a for some a>1. Show that f…

A: Let f:R--&gt;R and suppose that |f(x)-f(y)| less than or equal to |x-y|^a for some a&gt;1. Show that…

Q: A continuous function f, defined for all æ, has the following properties: 1. f is decreasing 2. f is…

A:

Q: For a function f: A --> B and subsets C and D of A and E and F of B, prove the following: f-1(E -…

A:

Q: Assume that p is a function that is defined for x &gt; 0 and satisfies p(a/b) = p(b) - p(a). Prove…

A: pab=p(b)-p(a)f(x)=p2-x2+x

Q: Draw a function f(x), defined for all real numbers x, with one local maximum and two local minimum.

A:

Q: Let F be the collection of all functions f: N-→ {0,1}. Is F countable ?

A: A set S  is countable if there exists a one-to-one map from the set to the set of all-natural…

Q: There exists a function f such that f(x) > 0, f(x) 0 for all x. True False

A: We are entitled to solve only one question at a time so, I am providing you an answer of the same.…

Q: If f(a) > f(x) in an open interval containing a, x + a, then the function value f(a) is a relative .…

A: Given: If f(a)&gt;f(x) in an open interval containing a, x≠a, then the function value f(a) is a…

Q: If f(a) &gt; f(x) in an open interval containing a, x ≠ a, then the function value f(a) is a…

A: Definition:

Q: Suppose that g is an even function of x and h is an odd function of x. Show that if g(x) + h(x) = 0…

A: Consider that g(x) is the even function. Then, g(-x)=g(x) for all the value of x. Consider that h(x)…

Q: (a) Assume that f(x) is a function defined by f(x) = x²-3x+1 2x - 1 for 2 ≤ x ≤ 3. Prove that f(x)…

A: Given.

Q: 2. Suppose f : X →Y and g : Y → Z are functions, and g of :X → Z is injective. (a) Show that f is…

A:

Q: (a) Let f: [a, b] → R be continuous and suppose that f(x) > 0 for all z. Show that there is some L>0…

A: As per the question (a), we are given a contentious positive function f : [a, b] → R. And we have to…

Q: Find all the strictly monotonic functions f: R→R such that f(x + f(y)) = f(x) + y, for all z, y € R.…

A: a. Consider the given condition, fx+fy=fx+y, for all x,y∈ℝ. Here we need to find the strictly…

Q: A function f(x) is said to have a removable discontinuity at `x-a` if: 1. ` f' is either not defined…

A:

Q: A function for which f(x) = f( - x) for all x in the domain of f is called a(n) ____function.

A: The types of functions help to fill the given blank. The definition that consists of the terms f(x)…

Q: Prove that if f and g are even functions, then fg is also an even function.

A: Given:If f and g are even functionthen fg is also an even function

Q: (b) Let X = (a, b]. Construct a continuous function f : X → R (set of real numbers) which is…

A:

Q: State the Mean-value Theorem, and use it to prove that a) if f(x) is differentiable and f'(x) > 0…

A:

Q: Prove that f(x) = (x – 1)2 + (x + 1)² is an even function. A function k(x) is an even function if…

A:

Q: Assume that f(x) is a function defined by x²-3x+1 f(x) = 2x - 1 for 2 ≤ x ≤ 3. Prove that f(x) is…

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: Prove that f(x) = (x- 1)2 + (x + 1)? is an even function. A function k(x) is an even function if…

A:

Q: 1. Consider the functions f : A → B and g : B → C. a. If f and g are both injective, show that so is…

A:

Q: Suppose that F is an increasing function and F(0) > 0. Define G(x) = [ F. Prove that G is…

A: Consider the given function is increasing then, F(0)&gt;0 Given integral is G(x)=∫0xF(x)dx. Then,…

Q: Let f (x) = 2x3 - 3x2 - 36x + 14. Also, let a = 0 and b = 1. (a) Show that the function f is…

A:

Q: Let f be a function from R - {1} to R given by f(x) = x/(x-1). Then f is surjective; injective;…

A: Given function is f(x) = xx-1

Q: Rolle's theorem states that if a function f is continuous on the closed interval [a, b] and…

A: Solution; Given data, function f is continuous on the closed interval [a, b] and differentiable on…

Q: If f is C2 on an interval prove that f(2 + k) – 2/(2) + {(z =A) = r"(2). - lim h-0 h2 = f"(x). The…

A:

Q: Let f:Z→Z be defined by f(n) = 2n+ 1 for each n∈Z and g:R→R defined by g(x) = 2x+ 1 for each x∈R.…

A: Given function is Let f:Z→Z be defined by f(n) = 2n+ 1 for each n∈Z and g:R→R defined by g(x) = 2x+…

Q: Assume that g(x) is a function given by 2.x? + 3x +2 g(x) = x² + 2 Prove that g(x) is bounded for…

A:

Q: Define g(4) in a way that extends g(x) = (x2 - 16)/(x2 - 3x - 4) to be continuous at x = 4.

A: On simplification, we get

Q: For a function g(x), if there exists x E [a,b] such that g'(x) > 1, then we conclude that g(x) has…

A: Solution: Given, gx is a function and there exists x∈a,b such that g'x&gt;1. Then, gx has not a…

Q: Suppose g is a function continuous at c and g(c) > 0. Prove that there exists δ > 0 such that g(x) >…

A:

Q: Let g : Z x Z -→ Z x Z be the function given by g(x, y) = (3x + y, x – y). Find, with proof, an…

A:

Q: Let g(x) be a function defined on R such that g''(x)= 1/(1+x^2), g (0) = 0 and g'(0) = 0. Use the…

A: We are given that g''(x)= 1/(1+x2), g (0) = 0 and g'(0) = 0. So, g'(x) = tan-1(x) + c.  But g'(0) =…

Q: Prove that f(x) = (x – 1)2 + (x + 1)² is an even function. A function k(x) is an even function if…

A:

Q: Determine whether the function f (x) = {: —х in — 1 <x<0 in 0<x < 1 is even or odd,

A:

Q: Let f : R → R be an increasing function. Show that f(x) = x, ∀x ∈ R ⇔ f(x) = x, ∀x ∈ Q.

A:

Q: Suppose g is a function continuous at c and g(c) > 0. Prove that there exists δ > 0 such that g(x) >…

A: Let ε &gt; 0 such that g(c) − ε &gt; 0 (for instance one can take ε = g(c)/2).

Q: If f(a)&gt; f(x) in an open interval containing a, x ≠ a, then the function value f(a) is a…

A: If f(a)&gt; f(x) in an open interval containing a, x ≠ a, then the function value f(a) is a relative…

Q: Let n be a positive integer and let f: [0..n] - (0..n] be an injective function. Define the function…

A:

Q: Prove that f(x) = (x - 1) + (x+1) is an even function. A function k(x) is an even function if…

A:

Q: Define g(3) in a way that extends g(x) = (x2 - 9)/(x - 3) to be continuous at x = 3.

A: Here, the given function The function gx is continuous at  x=a, if the following holds: (i) ga is…

Q: Show that the functions(a) f(x)=1/(1+x2),x∈R, (b) g(x)=x/(1+x2),x∈R, are uniformly continuous, while…

A: Answer

Q: 6. Define functions f: A B and g: B- C. Prove that if f and g are both injective, then their…

A: Injective function: A function f:A→B is said to be injective, if f(x)=f(y)⇒x=y. Given: f:A→B and g:…