**Question 4.** For each of the following functions, determine whether it is injective. Provide a formal detailed proof to justify your answer.(a) $ f : \mathbb{N} \to \mathbb{Z} $ with the assignment rule $ f(x) = 2x^2 $.(b) $ H : \mathbb{P}_2 \to \mathbb{R} $ with the assignment rule $ H(p) = p'(0) + 2 $. (Clarification: Here $ p $ is a polynomial, and $ p'(x) $ is the derivative of $ p $, which is also a polynomial. Then $ p'(0) $ is a number, which you get from plugging in $ x = 0 $ into the polynomial $ p'(x) $.)(c) $ q : \mathbb{R} \to \mathbb{R} \times \mathbb{R} $ with the assignment rule $ q(x) = (x + 1, x^2) $.

(a) to prove injective if f(x1)=f(x2)⇒x1=x22x12=2x22⇒x12=x22⇒x1=±x2x1= - x2 it is not possible…

Answered: Question 4. For each of the following…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Concept explainers

Question

Expert Solution

Step 1

$(a) t o p r o v e i n j e c t i v e i f f (x_{1}) = f (x_{2}) \Rightarrow x_{1} = x_{2} 2 {x_{1}}^{2} = 2 {x_{2}}^{2} \Rightarrow {x_{1}}^{2} = {x_{2}}^{2} \Rightarrow x_{1} = \pm x_{2} x_{1} = - x_{2} i t i s n o t p o s s i b l e b e c a u s e d o m a i n o f f u c t i o n i s N h e n c e x_{1} = x_{2} t h e r e f o r e f (x) i s i n j e c t i v e$

Step by step

Solved in 2 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Single Variable$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions