(b) Let f : R³ →→ R' be a linear function. Two properties define a linear function: f(x) + f(y) = f(x +y) and f(rx) = rf(x). Using this information, show that f(x) = a'x for all x E R°, where a; = f(e;) for i = 1, 2, 3 and %3D ei = and e3 (c) Use at least one of the criteria of a linear function shown in question 2 (b) to conclude whether f(x) and w(x) are linear or not.
(b) Let f : R³ →→ R' be a linear function. Two properties define a linear function: f(x) + f(y) = f(x +y) and f(rx) = rf(x). Using this information, show that f(x) = a'x for all x E R°, where a; = f(e;) for i = 1, 2, 3 and %3D ei = and e3 (c) Use at least one of the criteria of a linear function shown in question 2 (b) to conclude whether f(x) and w(x) are linear or not.
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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Any help with b and c please
Transcribed Image Text:Use the following functions of x E R' to answer the rest of this
question:
f (x)
) = =; 9 (x) =; h (x) = 3x; w (x) = In(x)
%3D
(a) Which of the following are continuous? Also state the domain
of each, and whether the relevant function is one-to-one.
(i) h (х) + w (х)
(ii) ƒ (x) o g (x)
(b) Let f : R³ –→ R' be a linear function. Two properties define a
linear function: f(x)+ f(y)= f(x+y) and f(rx) = rf(x).
Using this information, show that f(x) = a'x for all x € R³,
f(e;) for i = 1,2,3 and
where a; =
ej =
and e3 =
(c) Use at least one of the criteria of a linear function shown in
question 2 (b) to conclude whether f(x) and w(x) are linear
or not.
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