Question #45759

a) Which of the following functions are 1-1 and which are onto? Justify your answer.
i) f : R ! R
0
given by f (x) = x
2
where R
0
is the set fx 2 Rjx 0g.
ii) f : R ! R given by f (x) = x
2
+ x + 1. (3)
b) If we consider the expression
1
2 3x
as a function on R, what will be its domain and
range? Will it have an inverse? Justify your answer. (2)
c) Let a =
(
1
2
p
2
;
p
3
2
p
2
;
1
p
2
)
and b =
(
1
p
2
; 0;
1
p
2
)
.
i) Find the direct cosines of a and b.
ii) Find the angle between a and b. (2)
d) Check that the vectors u =
(
3
5
;
4
5
; 0
)
, v =
(
4
5
;
3
5
; 0
)
and w = (0; 0; 1) are orthonormal.
Further, write the vector a = (1; 1; 2) as a linear combination of the vectors.

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