Solve You have given a function A: R → R with the following properties (x E R, n E N):X(n) = 0 , A(x + 1) = X(x) , A (n+Find two functions p, q : R → R with q(x) # 0 for all x such that X(x) = q(x)(p(x)+1).

Given: The function λ:R→R with the following properties, λn=0, λx+1=λx, λn+12=1.

Answered: You have given a function A : R → R…

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter3: Functions

Section3.3: Rates Of Change And Behavior Of Graphs

Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...

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You have given a function A: R → R with the following properties (x E R, n E N):
X(n) = 0 , A(x + 1) = X(x) , A (n+
Find two functions p, q : R → R with q(x) # 0 for all x such that X(x) = q(x)(p(x)+1).

Expert Solution

Step 1

Given:

The function $λ : R \to R$ with the following properties,

$λ (n) = 0, λ (x + 1) = λ (x), λ (n + \frac{1}{2}) = 1$ .

Step 2

Explanation:

obtain the function $p, q : R \to R$ with $q (x) \neq 0 \forall x \in R$ , such that $λ (x) = q (x) (p (x) + 1)$ .

That is, for any function $λ : R \to R$ there are function $p, q : R \to R$ with $q (x) \neq 0 \forall x \in R$ such that

$λ (x) = q (x) (p (x) + 1)$ .

Define,

$p (x) = \{\begin{matrix} 0 & i f x \in R w i t h λ (x) \neq 0 \\ - 1 & i f x \in R w i t h λ (x) = 0 \end{matrix}\}$ and

$q (x) = \{\begin{matrix} λ (x) & i f x \in R w i t h λ (x) \neq 0 \\ 1 & i f x \in R w i t h λ (x) = 0 \end{matrix}\}$

Then $q (x) \neq 0, f o r a l l x \in R$ .

Also, if $x \in R$ with $λ (x) \neq 0,$ $q (x) (p (x) + 1) = λ (x) (0 + 1) = λ (x)$ and

if $x \in R$ with $λ (x) = 0,$ $q (x) (p (x) + 1) = 1 (- 1 + 1) = 0 = λ (x)$ .

Therefore, the above defined function $p, q : R \to R$ works for the requirement that $q (x) (p (x) + 1) = λ (x)$ for all $\forall x \in R$ .

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You have given a function A : R → R with the following properties (x E R, n E N): A(n) = 0 , A(x+1) = X(x) , A (n+ = 1 Find two functions p, q : R → R with q(x) # 0 for all x such that X(x) = q(x)(p(x) + 1).