(i) Show that F(S) is an infinite set, and that f is injective but not surjective. (i) Is F(S) uncountable? Justify your answer.

Q: if and only if F-1: R(F) →X is bounded.

A:

Q: Find the interval on which f is increasing and in which interval f is decreasing

A:

Q: 9.J. If F is a non-empty closed set in R and if x ¢ F, is there a unique point of F that is nearest…

A: As per BARTLEBY GUIDELINES, I answered one question (Q-9.J.) and repost other questions separately.…

Q: If f is even, then f' is even.

A: The question is taken from the function in which we have to do that f is even then its derivative is…

Q: Let f(x)=6x^3-7x^2/cuberoot x^4. Evaluate f'(x) at x=2. f'(2)=

A: We have to find derivative of f(x) at x=2, where f(x) is given by

Q: F is algebraically closed.

A: (a) =) (b) Let f(x) be a irreducible polynomial in F[x].

Q: Let f be defined in an open interval that contains a=3 and assume that all x,…

A: f be defined in an open interval that contains a=3 and assume that all x,2x+6≤f(x)≤x2-4x+15

Q: Let f(x)=x -12x-1 on [-5,4]. F the Closed Interval Method to find t

A: fx=x3-12x-1f'x=3x2-12Critical point satisfy the condition f'x=03x2-12=03x2=12x2=123x2=4x=±2critical…

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: How would you "remove the discontinuity" of f? In other words, how would you define f(4) in order to…

A: Discontinuity of f can be removed by factorisation of the numerator

Q: The domain of f/g consists of numbers x for which g(x) __________  0 that are in the domains of both…

A: we have to fill the following blanks ;  The domain of f/g consists of numbers x for which g(x)…

Q: Let f(x) = (|x| + x2 − 1     if x ≤ 0          (|x| − x2 − 1     if x > 0 Is there some x in (−1, 1)…

A:

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: If F is continuous on the set of real numbers, then A True B) False

A: The solution is given in the next step.

Q: Compute f '(a) algebraically for a = 3. f(x) = 2x^2+ x

A: Solution: The objective is to find f '(a) algebraically for a = 3

Q: Explain, using Theorem 4, 5, 7 and 9, why the function is continuous at every number in its domain.…

A:

Q: For the function f: Z Z prove or disprove whether f is injective and/or surjective f(x) = 9x + 1

A: A detailed step by step solution is provided below.

Q: 14. Determine the points at which f is discontinuous. Identify the type of each discontinuity as…

A:

Q: . Show that the function f: R→ R defined as f(x) = 13x − 15 is one-to-one.

A:

Q: Determine whether the statement is true or false. If f and g are discontinuous at x = c, then so isf…

A: 1. If f and g are discontinuous at x=c, then so f+g. 2. If fx is continuous at x=c, then so is fx

Q: Give an example of a function f : (X1, d1) → (X2, d2) that sends compact sets to compact sets, but…

A:

Q: 10) Determine the intervals of continuity. (a) f(x) = -3r² + 5x – 8 (b) f(x) = x2 – 9 V3-x (c) f(x)…

A: Sketching the graph:

Q: If f is an even function, then f −1 exists.

A: Consider an even function ,f(x)=x2 For,x=±2,we have f(±2)=4 Hence, given function is not…

Q: 11. Let f be a f maximum at I. f'(c II. If f' III. If f" (А) П only

A: Given : f has relative maximum at x=c Suppose f is an absolute function then f has relative maximum…

Q: d) Let XCR be a set and f: X→ X be a function. If X is compact and convex, then f admits a fixed…

A: d) If X is compact subset of R then X is closed and bounded. If X  is convex subset of R then X is…

Q: Use the graph of f to find (a) the largest open interval on which f is increasing and (b) the…

A: Observe the graph, as x approaches 3 from the left the value of the function decreases and as x…

Q: Use the graph of f to find (a) the largest open interval on which f is increasing and (b) the…

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: (b) Suppose f'(x) > 1 for all x > 0, and f(0) = 0. Show that f(x) > x for all x> 0.

A: We will be using mean value theorem

Q: 11. What property must f(x) have in order for f'(x) to exist?

A:

Q: Let f(x)  = 1- x2/3 . Show that f(-1) = f(1) but there isno number c in  (-1, 1)such that  f'(c) =0…

A:

Q: f is also constant

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: 23-24 How would you “remove the discontinuity" of f ? In other words, how would you define f(2) in…

A:

Q: If f(x) =x²-x over x-1 and g(x) =x is it true that f=g?

A:

Q: A function f is ________ when there exists a positive real number c such that f (t + c) = f(t)for…

A: We know that for a functionft, if there exist a positive real number c such that ft+c=ft, for all t…

Q: 4 -6 4/ -2 2 6.

A:

Q: Show that if f is both even and odd, then f(x) = 0 for every x in the domain of f. %3D

A: Use the defination of a function for to be odd and even.

Q: Find the interval on Which f` is increasing and the interval of which f`is decreasing

A:

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: a. Use the fact that every nonempty interval of real numbers contains both rational and irrational…

A:

Q: 3. Determine the value of a so that f(x) is continuous on the entire real number line when: f(x) =…

A:

Q: 2. If f : A → B and g : B→ C are injective functions, then gof: A → C is also injective.

A: We need to prove/disprove statenemt two.

Q: Let f(x) Then (f o g)(x) — 5а — 4 and д(x) — г* — 4+ 2. 4х + 2. = x - - (go f)(x)

A:

Q: 7. Determine the points at which f is discontinuous. Identify the type of each discontinuity as with…

A:

Q: If f(x) is a function that is defined for all x E R, then f(ƒ-'(x)) = = x. True

A: Let,  f(x) = a*x+b f-1(x) = (x-b)/a

Q: 3. Give an example of a function f.N→N which is surjective but not injective.

A:

Q: 1. Suppose that f2(x) is a measurable function. Is it true that f(r) is measurable? Prove, or give a…

A:

Q: 7. Determine whether the funcion is even, odd, or neither: f(x) = 1+x2

A:

Q: Let f defined by f(x) = 2x³ + 3x² − 5 at [-1,1], Consider the following statements 1-It reaches an…

A: The given function,   fx=2x3+3x2-5 at -1,1

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.