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.

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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