Thomas Calculus 13th [Solutions]

Section 1.2 Combining Functions; Shifting and Scaling Graphs 9
5. (a) 2 (b) 22 (c) x
(d) ( x
2
5) 3
2
x 10x 22 (e) 5 (f ) 2
(g) x 10
2
(h) ( x
2
3) 3
4
x
2
6x
6
2
2
6. (a)
1
3
(b) 2 (c)
1
x
1 1 x
x 1
(d) 1 (e) 0 (f ) 3 x
4
(g) x 2
(h)
1 1 x 1
1 x 2 x 2
x
1
1 x 1
7. ( f g h)( x) f ( g( h( x ))) f ( g(4 x )) f (3(4 x )) f (12 3 x) (12 3 x)
1 13 3x
8.
2
( f g h)( x) f ( g( h( x))) f ( g( x ))
2
f (2( x ) 1)
2 2
f (2x 1) 3(2x
1)
2
4 6x
1
9. ( f g h)( x) f ( g( h( x)))
f g
1
f 1
x
1
4
x
f
x x 5x
1
1
1 4x 1 4x 1 4x
10. ( f g h)( x) f ( g( h( x))) f g 2 x
f
2 2 x
3 x
f
2 x 2
3
3
2 x 2 x 2 8 3x
3 x 7 2x
2 x 1
x
11. (a) ( f g)( x )
(b) ( j g)( x )
(c) ( g g)( x)
(d) ( j j)( x )
(e) ( g h f )( x )
(f ) ( h j f )( x)
12. (a) ( f j)( x )
(b) ( g h)( x )
(c) ( h h)( x)
(d) ( f f )( x )
(e) ( j g f )( x )
(f ) ( g f h)( x)
13. g(x) f (x) ( f g)( x)
(a) x 7 x x 7
(b) x 2 3x 3( x 2) 3x
6
(c) 2
x x 5
2
x 5
(d) x
x
x
x 1 x 1
x 1 x
1 x ( x 1)
(e) 1
x 1
(f ) 1
x
x
x
1 1 x
x
1
x
x
1
x
14. (a) ( f g)( x) | g( x)| 1 .
x 1
( ) 1
(b) ( f g )( x g x
)
x
1
g( x) x 1
1 x
g( x) x 1
2
(c) Since ( f g)( x) g( x) | x|,
g( x) x .
(d) Since ( f g)( x) f x | x|,
1 x
x
1
1 g( x)
1 1
x 1 g( x) ,
so g( x) x 1.
2
f ( x) x . (Note that the domain of the composite is [0, ).)
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