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Chapter 1 Functions Exam 1.If y = x2 + 8x - 33, the values of x for which y = 0 are Ans: x = 3 and x = -11 x6 x , find g(x + 3). x+9 Ans: x + 3

g ( x) 

2. If

3. Find the natural domain and range for f ( x)  3x  9 .  3,   Ans: Domain: x  3 or 0,   Range: y  0 or  4.Find the natural domain and range for 8x  3 f ( x)  5x  8 . 8  8   8 x  ,      ,   5  5  5 or  Ans: Domain: y

Range:

8 5 or

8 8    ,    ,   5 5  

5.Find the natural domain for

f ( x)  Ans: x  2  2,   or

4 x2 .

6..Find the natural domain of

x3 x 1 . x 1

f ( x) 

Ans: x  3 or  ,3  1,   or

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2 7. Find the natural domain for h( x)  70  9 x  x . Ans: [-14, 5]

8. Find the natural domain for 6x f ( x)  x  11 . Ans: x  11 or x  0 or

 , 11  0,  

9. Find f(8) if f(x) = x2 - 5x + 11. Ans: 35

10. What is the natural domain of the function –7 f ( x)  6 2  x  1 ? Ans: x  1  x4 3x g ( x)  f    3  , write an expression for g(x) and find its range x  4 and 11. If and domain. 3x  12 g ( x)  x  16 Ans:

f ( x) 

Domain: Range:

x  16 y3

or or

 , 16   16,    ,3   3,  

x2  4 x 1 f ( x)  3x 12. Let and g(x) = x - 2, find (f + g)(x). 2 4 x  10 x  1 ( f  g )( x)  3x Ans:

3x 2  5 x  1 2x 13. Let and g(x) = x - 5, find (f - g)(x). 2 x  5x  1 ( f  g )( x)  2x Ans: f ( x) 

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 f  2 x 2  4 x  48   x 7 x 14. If and g(x) = x - 6, find  g  . 2x  8 Ans: 7 x for x  6 f ( x) 

2x 8  x 2 and g ( x)  7 x , find f  g ( x) . 15. Let 2 7x Ans: 8  7 x f ( x) 

16. Let g(x) = 5x + 8 and h(x) = 3x - 7, find Ans: 15t + 48 domain:

g  h  t  5

all real numbers

and its domain or

17. Let f ( x)  x  8 and g ( x)  3 x , find g  f ( x) and its domain. 4 Ans: 3 x  8 Domain:

x8

or

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8,  

 ,  


2 18. Express h( x)  x  1 as the composition of two functions such that h(x) = f  g(x). Ans: f ( x)  x

g ( x)  x 2  1

2 19. Sketch the graph of f ( x)  16  6 x  x by completing the square. Ans:

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20. f(x) = x2 - 7 and g(x) = x4 + 5. Find f(g(x)). Ans: x8 + 10x4 + 18 21. Is f(x) = (x + 11)3 an even function, an odd function, or neither? Ans: neither 22. Determine whether the graph y = –3x2 - 6 is symmetric about the x-axis, the y-axis, or the origin. Ans: Symmetric about the y-axis 23. Determine whether the graph y = –3x3 - 8x is symmetric about the x-axis, the y-axis, or the origin. Ans: The graph is symmetric about the origin. 24. Determine whether the graph x5 = –7y5 - 5y is symmetric about the x-axis, the y-axis, or the origin. Ans: The graph is symmetric about the origin. 25. Find all intercepts of 4x2-y2 = 20 and determine symmetry about the x-axis, the y-axis, or the origin. Ans: x-intercepts: x   5 y-intercepts: none The graph is symmetric about both the x-axis and y-axis, it is symmetric about the origin.

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y

–9 8 x  x3 and determine symmetry about the x-axis, the y-axis,

26. Find all intercepts of or the origin. Ans: x-intercepts: none y-intercepts: none The graph is symmetric about the origin.

27. Find all intercepts of x = 5y4 - 25y2 and determine symmetry about the x-axis, the y-axis, or the origin. Ans: x-intercept: x = 0 y-intercept: y   5 and y = 0 The graph is symmetric about the x-axis. 28. Find all intercepts of y4 = |2x| + 4 and determine symmetry about the x-axis, the y-axis, or the origin. Ans: x-intercept: none 4 y-intercept: y   4 The graph is symmetric about both the x and y-axes. The graph is symmetric about the origin. 29. Find all intercepts of y9 = |9x| - 7 and determine symmetry about the x-axis, the y-axis, or the origin. 7 x 9 Ans: x-intercept: 9 y-intercept: y  7 The graph is symmetric about the y-axis.

30. Given f(x) = x2 + 14x + 40 is not a one to one function. Modify the domain of f so that it will be a one-to-one function.  , –7 or on the range  –7,   Ans: f ( x) is one-to-one on the range

2 31 Given that f ( x)  9  x is not a one-to-one function. Modify the domain of f so that it will be a one-to-one function. Ans: f(x) is one-to-one on [-3, 0] or on [0, 3]

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32. Find f -1(2) if

f ( x)  –11x5  5

Ans: f -1(2) =

7 8 .

9 88 f ( x) 

10 x  7 2 x  10 . What does this result tell you about the graph of f?

33. Find the inverse of 10 x  7 f 1 ( x)  2 x  10 Ans: The graph of f is symmetric about the line y = x. 4 34. Find f -1(x) if f ( x)  8x  5 . x4  5 f 1 ( x)  8 Ans:

35. Determine whether or not f(x) = 2x5+9x3 + 6x - 7 is a one-to-one function. Ans: Yes, f(x) is a one -to-one function.

g ( x) 

36. Determine if f(x) = 9x + 2 and Ans: Yes

x2 9 are inverses of each other.

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