Question 7 Consider the set L of all liner functions on R. These are h(x) = ax + b with a € R and b e R. Define the addition + as usual: (f + g)(x) = f(x) + g(x). Define the multiplication as composition of functions: (fo 9)(x) = f(g(x)). Is L with these operations + and o a ring? Explain your answer.
Question 7 Consider the set L of all liner functions on R. These are h(x) = ax + b with a € R and b e R. Define the addition + as usual: (f + g)(x) = f(x) + g(x). Define the multiplication as composition of functions: (fo 9)(x) = f(g(x)). Is L with these operations + and o a ring? Explain your answer.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Question 7**
Consider the set \( \mathcal{L} \) of all linear functions on \( \mathbb{R} \). These are defined as:
\[ h(x) = ax + b \]
where \( a \in \mathbb{R} \) and \( b \in \mathbb{R} \).
Define the addition \( + \) as usual:
\[ (f + g)(x) = f(x) + g(x). \]
Define the multiplication as the composition of functions:
\[ (f \circ g)(x) = f(g(x)). \]
Is \( \mathcal{L} \) with these operations \( + \) and \( \circ \) a ring? Explain your answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7071178f-afef-4ac0-98b1-2b0472d13bc3%2F8835f316-8675-41b9-b04f-7ce0006f0226%2Fyh68p8p_processed.png&w=3840&q=75)
Transcribed Image Text:**Question 7**
Consider the set \( \mathcal{L} \) of all linear functions on \( \mathbb{R} \). These are defined as:
\[ h(x) = ax + b \]
where \( a \in \mathbb{R} \) and \( b \in \mathbb{R} \).
Define the addition \( + \) as usual:
\[ (f + g)(x) = f(x) + g(x). \]
Define the multiplication as the composition of functions:
\[ (f \circ g)(x) = f(g(x)). \]
Is \( \mathcal{L} \) with these operations \( + \) and \( \circ \) a ring? Explain your answer.
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