Understanding Engineering Mathematics

3.3.4 Odd and even functions

(i) odd (ii) even (iii) odd

(iv) neither (v) even (vi) even

(vii) odd (viii) neither

3.3.5 Composition of functions

f(g(x))=

x^2 + 1

(x^2 + 1 )− 1

=

x^2 + 1

x^2

g(f(x))=

(

x

x− 1

) 2

+ 1 =

2 x^2 − 2 x+ 1

(x− 1 )^2

3.3.6 Inequalities

(i) x>^52 (ii) − 3 <x< 3 (iii) x≥2,x≤− 1

(iv) − 3 <x<1(v)−^12 ≤x≤^32 (vi) x>^92 ,x<^72

(vii)^23 <x< 1

3.3.7 Inverse of a function

(i)

x− 1

2

domain: all values; range: all values

(ii)

1 + 2 x

1 −x

domain: all values=1; range: all values=− 2

(iii)

√

x−1 domain: any value≥1; range: any value≥ 0

3.3.8 Series and sigma notation

A. (i)

∑^5

n= 1

n^3 (ii)

∑^33

n= 1

3 n (iii)

∑^50

n= 2

1

n

(iv)

∑∞

n= 0

(− 1 )n

1

3 n

Note that alternative forms are permissible – for example an equally acceptable answer

to (iii) is

∑^49

n= 1

1

n+ 1

.

B. (i) 1+

1

2

+

1

3

+

1

4

(ii) 1+ 1 + 2 +6 (note 0!= 1 )

(iii)

1

2

+

1

6

+

1

12

+

1

20