Prove the following statement. If f: R→ R is onto and c is any nonzero real number, then (c f) is also onto. Proof: Suppose f: R→ R is onto and c is any nonzero real number. Given any real number z in the ---Select--- ✓ of f, we must show that there is a real number in the -Select--- ✓ of f, say x, such that (c f)(x) = z. Let x be such that f(x) = and since ---Select--- ✓ this is a real number. So, since f is ---Select--- ✓, ✗ is in the domain of f. Now (c. f)(x) = = C • = Z. f(x) by definition of composition because f(x) = Thus, (c f) is onto [as was to be shown].
Prove the following statement. If f: R→ R is onto and c is any nonzero real number, then (c f) is also onto. Proof: Suppose f: R→ R is onto and c is any nonzero real number. Given any real number z in the ---Select--- ✓ of f, we must show that there is a real number in the -Select--- ✓ of f, say x, such that (c f)(x) = z. Let x be such that f(x) = and since ---Select--- ✓ this is a real number. So, since f is ---Select--- ✓, ✗ is in the domain of f. Now (c. f)(x) = = C • = Z. f(x) by definition of composition because f(x) = Thus, (c f) is onto [as was to be shown].
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 6TFE: Label each of the following statements as either true or false. Let R be a relation on a nonempty...
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![Prove the following statement.
If f: R→ R is onto and c is any nonzero real number, then (c f) is also onto.
Proof: Suppose f: R→ R is onto and c is any nonzero real number.
Given any real number z in the ---Select--- ✓ of f, we must show that there is a real number in the -Select--- ✓ of f, say x, such that (c f)(x) = z. Let x be such that f(x) =
and since ---Select--- ✓
this is a real number. So, since f is ---Select--- ✓, ✗
is in the domain of f.
Now
(c. f)(x)
=
= C •
= Z.
f(x) by definition of composition
because f(x) =
Thus, (c f) is onto [as was to be shown].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff3cf874b-7a7b-478f-a7b8-421442e72224%2F50ae6f40-370c-4b4c-9e6e-1f105b5c312b%2F9t7fsl_processed.png&w=3840&q=75)
Transcribed Image Text:Prove the following statement.
If f: R→ R is onto and c is any nonzero real number, then (c f) is also onto.
Proof: Suppose f: R→ R is onto and c is any nonzero real number.
Given any real number z in the ---Select--- ✓ of f, we must show that there is a real number in the -Select--- ✓ of f, say x, such that (c f)(x) = z. Let x be such that f(x) =
and since ---Select--- ✓
this is a real number. So, since f is ---Select--- ✓, ✗
is in the domain of f.
Now
(c. f)(x)
=
= C •
= Z.
f(x) by definition of composition
because f(x) =
Thus, (c f) is onto [as was to be shown].
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