2. In each item below, you are given sets A and B, and a formula for a (non?) function f: A → B. In each case, determine whether or not the formula gives a well-defined function. Fully justify your answers. (a) A = Q = B, f(r) = a root in B of the polynomial X²-r (b) A=C=B, f(2)= a root in B of the polynomial X² - z (c) A = P(R) - {0}¹, B = R, ƒ(E)= the smallest element of E (d) A = P(R), B = R, f(E)= the smallest element of E if E admits a smallest ele- ment, and f(E)= 0 otherwise A function f A → B is injective (or one-to-one) if it satisfies the following condition: for all a₁, a₂ € A, if f(a₁) = f(a₂), then a₁ = a2; f is surjective (or onto) if f(A) = B, or, more explicitly, if for each b E B there exists a € A such that f(a) = b; f is bijective if f is both injective and surjective. Here are some examples. • The squaring function f: RR given by f(x) = x² is not injective because f(-1) = 1 = f(1); f also fails to be surjective because -1 is not in the range of f. The function g: R→ [0,00) given by g(x) = x² is not injective for the same reason that the function f of the previous example is not injective, since g(-1) = 1 = g(1); but g is surjective, because every nonnegative real number admits a square root (this is a result in introductory real analysis). • The sine function sin: R R is very far from injective: it is 27-periodic, and assumes all of its values infinitely often. It is not surjective either, because its range is [-1,1]. • If we define S: [-π/2, π/2] → [-1, 1] by S(x) = sin(x), then methods from calculus show that S is both injective and surjective, i.e., S is bijective. Restricting the domain and codomain of sin in this way is how one obtains the arcsin function, which is the inverse of the function we have called S (see below for more on inverse functions). • The swapping function from Rx R to itself sending (x, y) to (y, z) is bijective. (Verify this!) • The exponential function exp: R→ (0, ∞) is bijective. (I write exp(r) for what is typically written e in calculus textbooks. This notation is pretty standard outside

2. In each item below, you are given sets A and B, and a formula for a (non?) function f: A → B. In each case, determine whether or not the formula gives a well-defined function. Fully justify your answers. (a) A = Q = B, f(r) = a root in B of the polynomial X²-r (b) A=C=B, f(2)= a root in B of the polynomial X² - z (c) A = P(R) - {0}¹, B = R, ƒ(E)= the smallest element of E (d) A = P(R), B = R, f(E)= the smallest element of E if E admits a smallest ele- ment, and f(E)= 0 otherwise A function f A → B is injective (or one-to-one) if it satisfies the following condition: for all a₁, a₂ € A, if f(a₁) = f(a₂), then a₁ = a2; f is surjective (or onto) if f(A) = B, or, more explicitly, if for each b E B there exists a € A such that f(a) = b; f is bijective if f is both injective and surjective. Here are some examples. • The squaring function f: RR given by f(x) = x² is not injective because f(-1) = 1 = f(1); f also fails to be surjective because -1 is not in the range of f. The function g: R→ [0,00) given by g(x) = x² is not injective for the same reason that the function f of the previous example is not injective, since g(-1) = 1 = g(1); but g is surjective, because every nonnegative real number admits a square root (this is a result in introductory real analysis). • The sine function sin: R R is very far from injective: it is 27-periodic, and assumes all of its values infinitely often. It is not surjective either, because its range is [-1,1]. • If we define S: [-π/2, π/2] → [-1, 1] by S(x) = sin(x), then methods from calculus show that S is both injective and surjective, i.e., S is bijective. Restricting the domain and codomain of sin in this way is how one obtains the arcsin function, which is the inverse of the function we have called S (see below for more on inverse functions). • The swapping function from Rx R to itself sending (x, y) to (y, z) is bijective. (Verify this!) • The exponential function exp: R→ (0, ∞) is bijective. (I write exp(r) for what is typically written e in calculus textbooks. This notation is pretty standard outside

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section4.1: Polynomial Functions Of Degree Greater Than

Problem 49E

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