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MATH 153 TEST 3 (Sample, Key) NAME:
Sections: 3.4 - 3.8, 4.
1. a) 4 a + 2 h โ 4 b) โ 6 a โ 3 h + 2 2. Find the domain of:
a) [ 2 , 5 )U ( 5 ,โ) b) [ 0 , 5 )
c) [ 0 , 1 )U [ 2 ,โ)(use sign diagram covered in section 2.7) d) [ โ 3 ,โ 1 )U (โ 1 , 1 )U( 1 , 4 ]
3. Determine whether f is even , odd, or neither:
a) even b) neither c) odd
4. Sketch the graph of f for the given value of b and c : f ( x )=| x โ c |+ b
a) c = -2, b = 1
b) c = 3, b = -
x^ x
5. The graph of a function f is shown in the figure.
Sketch the graph of the given equation:
a) f ( x + 5 )
b) f ( x )โ 5
c) โ f ( x + 5 )+ 2
d) f ( x โ 2 )โ 3
e) 2 f ( x )
f) ( ) 2
โ f x f
e
d
c
a
b
12. Maximum area is 1800, the two sides are 30 by 60.
13. a)
4
x
x x g
f b) [ โ 2 , 0 )U( 0 ,โ)
14. a) 1122 b) 97
15. a) ( f o g )( x )= x ; domain R โ{ 3 }
b) ( g o f )( x )= x ; domain R โ{ 1 }
16. a)
3 2
x
x f o g x ; domain , 2 } 3
R โ{
b) x
x g f x 3
o =โ ; domain , 0 } 2
R โ{โ
17. a) ( f o g )( x )= x + 5 โ 3 ; domain [ โ 4 ,โ)
b) ( g o f )( x )= x โ 3 + 5 ; domain [ 3 ,โ)
18. a) ( f o g )( x )= โ 1 โ x ; domain ( โโ, โ 1 ] (see example 11 in 3.7 handout for this problem)
b) ( )() 3 4
2 g o f x = โ x โ ; domain [ โ 13 ,โ 2 ]U[ 2 , 13 ]
19. a) f x = x + x
3 ( ) Yes, but f x = x โ x
3 ( ) is not. Use Graphmatica and graph both functions to see why.
b) No c) Yes d) No
20. a)
2 3
โ
x
x f b)^3
1
4
5 x f
โ c)
1 5 = 2 +( โ 1 )
โ f x
21. a) -1 b) 4 c) 5 22. a) f ( x ) > 0 (above) when -3 < x < 1/2 or x > 3, f ( x ) < 0 (below) when x < -3 or 1/2 < x < 3
b) f ( x ) > 0 (above) when x > 1, f ( x ) < 0 (below) when x < -1 or -1 < x < 1
c) f ( x ) > 0 (above) when x < -2 or x > 2, f ( x ) < 0 (below) when -2 < x < 2
Note: It is very helpful to see the graph of each using Graphmatica
Bonus:
a) ( f o g )( x )= ( x โ 15 )+ 2 โ
x โ 15 ; domain [ 15 ,โ)
b) ( )() 2 15 ( 5 )( 3 )
2 g o f x = x + x โ = x + x โ ; domain [ โโ, โ 5 ]U[ 3 ,โ)
(Hint: it is helpful you see the solution of problem 21 in section 3.7)
