Let f : R → R , f ( x ) = x^2 ( x −...
Let f : R → R , f ( x ) = x^2 ( x − 3).
(a) Given a real number b , find the number of elements in f ^(-1) [ { b } ]. (The answer will depend on b . It will be helpful to draw a rough graph of f , and you probably will need ideas from calculus to complete this exercise.)
(b) Find three intervals whose union is R , such that f is injective as a function on each interval.
(c) Use your answers to (a) and (b) to define the domain and codomain of three bijective functions.
Homework Answers
1. Let f: R\{-1} → R, f() = 2+1 (a) Prove that f is not an...
1. Let f: R\{-1} → R, f() = 2+1 (a) Prove that f is not an increasing function on its domain, but its restrictions to intervals fl-20.-1) and fl(-1,00) are strictly increasing. (b) Find a codomain for fl-1.00) that makes the function bijective. Find the composi- tional inverse of our function. Sketch both our function and its inverse on the same set of axes.
Consider the following functions, where I and J denote two subsets of the set R of...
Consider the following functions, where I and J denote two subsets of the set R of real numbers. f: R→R x→1/√(x+1) f(I,J): I→J x→ f(x) (a) What is the domain of definition of f? (b Let y be an element of the codomain of f. Solve the equation f(x)=y in x. Note that you may have to consider different cases, depending on y. (c) What is the range of f? (d) Is f total, surjective, injective, bijective? (e) Find a...
5. Let A = P(R). Define f : R → A by the formula f(x) =...
5. Let A = P(R). Define f : R → A by the formula f(x) = {y E RIy2 < x). (a) Find f(2). (b) Is f injective, surjective, both (bijective), or neither? Z given by f(u)n+l, ifn is even n - 3, if n is odd 6. Consider the function f : Z → Z given by f(n) = (a) Is f injective? Prove your answer. (b) Is f surjective? Prove your answer
Question 8 (6 marks) Consider the function f: [-2, 2] shown in the diagram. R whose...
Question 8 (6 marks) Consider the function f: [-2, 2] shown in the diagram. R whose graph is (a) State the largest intervals on which f is injective f(z) (b) For each interval I you found in (a), sketch the graph of the inverse g of the restriction of f to I and state the domain and range of g (c) If we know that f(-x) = f(x) for all r e dom(f), what can we conclude about the relationship...
Let X be a set with an equivalence relation ∼. Let f : X/ ∼→ Y...
Let X be a set with an equivalence relation ∼. Let f : X/ ∼→ Y be a function with domain as the quotient set X/ ∼ and codomain as some set Y . We define a function ˜f, called the lift of f, as follows: ˜f : X → Y, x 7→ f([x]). We define a function Φ : F(X/ ∼, Y ) → F(X, Y ), f 7→ ˜f. (1) Is Φ injective? Give a proof or a...
10.3.1 Exercises 10.33: An ordinary function graph combines domain and range information in a single picture: we plot the ordered pair (x, f(x)). several variables we quickly run out of pictures...
10.3.1 Exercises 10.33: An ordinary function graph combines domain and range information in a single picture: we plot the ordered pair (x, f(x)). several variables we quickly run out of pictures we can draw, but there is simple alternative, which we illustrate first with an ordinary function from R to R. (This is exactly a domain-range picture, introduced in Section 1.3.) Take f(x) 2. Draw the domain of the function as a single vertical line (fair: the domain is R)....
6. (Extra Credit) Let I be the interval (0,1). Define F(I) = {f:I+I:f is a function},...
6. (Extra Credit) Let I be the interval (0,1). Define F(I) = {f:I+I:f is a function}, the set of all functions from the interval (0,1) to itself. (a) Thinking about the graph of a function, define a one-to-one function F(1) ► PIXI). Prove your function is one-to-one (remember that functions fi and f2 are equal when they have the same domain and codomain, and fi(x) = f2(x) for every x in the domain). (b) Given a set A CI, define...
How do I prove this function is not surjective? 3.) Let f: R-R, f(x)-x2+ x+1 and...
How do I prove this function is not surjective?
3.) Let f: R-R, f(x)-x2+ x+1 and Show that f is not injective and not surjective Justify that g is bijective and find gt. PIR, Show all the wortky) Not Surtechive: fx) RB Surjective: ye(o,oo) hng (g) 8 gon)-es is bijecelive g(x)-ex+s
2 er Let I be an interval of R, and define the function f :I→ R by f(x) 1 +e2z or every z EZ. (a) Find the largest...
2 er Let I be an interval of R, and define the function f :I→ R by f(x) 1 +e2z or every z EZ. (a) Find the largest interval T where f is strictly increasing. (b) For this interval Z, determine the range f(T) (c) Let T- f(I). Show that the function f : I -» T is injective and surjective. (d) Determine the inverse function f-i : T → 1. (e) Verify that (fo f-1)()-y for every y E...
5. [2 pts] Consider a function f: Z → Z and f(x) = x'. Answer the...
5. [2 pts] Consider a function f: Z → Z and f(x) = x'. Answer the following questions. a. Is f(x) invertible? Justify your answer. b. Suppose that the domain and codomain change to real number, thus f:R → R. Then is f(x) invertible? Justify your answer.
1. Let f: R\{-1} → R, f() = 2+1 (a) Prove that f is not an increasing function on its domain, but its restrictions to intervals fl-20.-1) and fl(-1,00) are strictly increasing. (b) Find a codomain for fl-1.00) that makes the function bijective. Find the composi- tional inverse of our function. Sketch both our function and its inverse on the same set of axes.
5. Let A = P(R). Define f : R → A by the formula f(x) = {y E RIy2 < x). (a) Find f(2). (b) Is f injective, surjective, both (bijective), or neither? Z given by f(u)n+l, ifn is even n - 3, if n is odd 6. Consider the function f : Z → Z given by f(n) = (a) Is f injective? Prove your answer. (b) Is f surjective? Prove your answer
Question 8 (6 marks) Consider the function f: [-2, 2] shown in the diagram. R whose graph is (a) State the largest intervals on which f is injective f(z) (b) For each interval I you found in (a), sketch the graph of the inverse g of the restriction of f to I and state the domain and range of g (c) If we know that f(-x) = f(x) for all r e dom(f), what can we conclude about the relationship...
10.3.1 Exercises 10.33: An ordinary function graph combines domain and range information in a single picture: we plot the ordered pair (x, f(x)). several variables we quickly run out of pictures we can draw, but there is simple alternative, which we illustrate first with an ordinary function from R to R. (This is exactly a domain-range picture, introduced in Section 1.3.) Take f(x) 2. Draw the domain of the function as a single vertical line (fair: the domain is R)....
6. (Extra Credit) Let I be the interval (0,1). Define F(I) = {f:I+I:f is a function}, the set of all functions from the interval (0,1) to itself. (a) Thinking about the graph of a function, define a one-to-one function F(1) ► PIXI). Prove your function is one-to-one (remember that functions fi and f2 are equal when they have the same domain and codomain, and fi(x) = f2(x) for every x in the domain). (b) Given a set A CI, define...
How do I prove this function is not surjective?
3.) Let f: R-R, f(x)-x2+ x+1 and Show that f is not injective and not surjective Justify that g is bijective and find gt. PIR, Show all the wortky) Not Surtechive: fx) RB Surjective: ye(o,oo) hng (g) 8 gon)-es is bijecelive g(x)-ex+s
2 er Let I be an interval of R, and define the function f :I→ R by f(x) 1 +e2z or every z EZ. (a) Find the largest interval T where f is strictly increasing. (b) For this interval Z, determine the range f(T) (c) Let T- f(I). Show that the function f : I -» T is injective and surjective. (d) Determine the inverse function f-i : T → 1. (e) Verify that (fo f-1)()-y for every y E...
5. [2 pts] Consider a function f: Z → Z and f(x) = x'. Answer the following questions. a. Is f(x) invertible? Justify your answer. b. Suppose that the domain and codomain change to real number, thus f:R → R. Then is f(x) invertible? Justify your answer.
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