Define a function ?∶ ℝ→ℝ by ?(?)={?+1,[?] ?? ??? ?−1,[?]?? ???? where [x] is the integer...

Define a function ?∶ ℝ→ℝ by

?(?)={?+1,[?] ?? ??? ?−1,[?]?? ????

where [x] is the integer part function. Is ? injective?

(b) Verify if the following function is bijective. If it is bijective, determine its inverse.

?∶ ℝ\{5/4}→ℝ\{9/4} , ?(?)=(9∙?)/(4∙?−5)

Expert Solution

Colby Messinger answered 2 years ago

Consider the function ?: ℝ → ℝ defined by ?(?) = ? if ? ∈ ℚ...

Consider the function ?: ℝ → ℝ defined by ?(?) = ? if ? ∈ ℚ and ?(?) = ? 2 if ? ∈ ℝ ∖ ℚ. Find all points at which ? is continuous

Topology question: Show that a function f : ℝ → ℝ is continuous in the ε...

Topology question: Show that a function f : ℝ → ℝ is continuous in the ε − δ definition of continuity if and only if, for every x ∈ ℝ and every open set U containing f(x), there exists a neighborhood V of x such that f(V) ⊂ U.

Let f(x) = {(C/x^n if 1≤ x <∞; 0 elsewhere)} where n is an integer >1....

Let f(x) = {(C/x^n if 1≤ x <∞; 0 elsewhere)} where n is an integer >1. a. Find the value of the constant C (in terms of n) that makes this a probability density function. b. For what values of n does the expected value E(X) exist? Why? c. For what values of n does the variance var(X) exist? Why?

Define a function average_entries_for(integer) that takes in an integer than represents the number of times the...

Define a function average_entries_for(integer) that takes in an integer than represents the number of times the function will ask the user to input a number. The function will return the average of all the numbers entered. You must use a for loop to implement this function. (Hint: use range)

Show that "f(x) = x^3" is continuous on all of ℝ

Consider the function ?: (−1,1) × (−1,1) → ℝ given by ?(?, ?) = sin(?? +...

Consider the function ?: (−1,1) × (−1,1) → ℝ given by ?(?, ?) = sin(?? + ?? + ?2 ). 1. Find a bound for the directional derivative of ? in any direction, i.e. find a constant ? such that |???(?, ?)| ≤ ? for all (?, ?) ∈ (−1,1) × (−1,1) and ? ∈ ℝ 2 with |?| = 1.

C programming: Define a function computer_choose that chooses a random integer between 1 and 1000 (inclusive)....

C programming: Define a function computer_choose that chooses a random integer between 1 and 1000 (inclusive). Note: the tests call your function 1000 times to verify that no results are <0 or >1000. Rarely, this can score an incorrect function as correct.

Find the runtime of this function, where n is an integer. int function(int n) { int...

Find the runtime of this function, where n is an integer. int function(int n) { int a = 0; for (int i = n; i > 0; i--) { for (int j = 0; j < i; j++) { a = a + i + j; } } return a; } Find the runtime of this function, where m and n are two integers. int f(int m, int n) { if (m==1 || n==1) { return 1; } return f(m,...

Let f(x)= kx^k-x^(k-1)-x^(k-2)-...-x-1, where k is an integer greater than or equal to 1. Prove the...

Let f(x)= kx^k-x^(k-1)-x^(k-2)-...-x-1, where k is an integer greater than or equal to 1. Prove the roots of f have absolute value less than or equal to 1. This is possibly using Cauchy's Estimates for Roots of Polynomials or Ostrowski's Theorem, but I'm not sure how to use them.

Given that h(x) = x.sinx . Find the root of the function h(x) = 1, where...

Given that h(x) = x.sinx . Find the root of the function h(x) = 1, where x is between [0, 2] using substitution method.

Question