Review for test 1 (MW-TR)
1) Factor completely x3+ 7x2 – 4x - 28
2) Divide using long division.
( 12x3+6 x2 - 6x + 10) ÷ (3x2 + 5)
3) Find f(x + 1) when f(x) = 4x2 - 2x + 5.
4) Find f(2x) when f(x) = 2x2 - 3x + 2.
Identify the intervals where the function is changing as requested.
5) Decreasing
6) Increasing
7) Determine the end behavior of the following polynomial functions
a) f(x) = -3x5 + 4x4-7x3 -5x2 + 2x – 6
b) f(x) = -x(x-2)2(x + 3)3(x2 + 5)
8) Factor completely 64 x3- 1
9) Factor completely
125 x3+ 27
10) Analyze the graph of the function f(x) = 2(x+1)2 (x - 2)(x2 + 3) as follows
a) Determine the end behavior: Find the power function that the graph of f resembles for large values of x.
b)Find the x-intercepts
c) Determine whether the graph crosses or touches the x –axis at each x-intercept
d) Use the information in a)-c) to sketch the graph of f.
11) Use the graph of the function f, given below, to sketch the graph of the given function g(x) = -f(x + 2) +2
Use the transformations. Do one transformation at a time
12) Determine whether or not the graph is a graph of a function of x. Explain.
13) Determine whether or not the graph is a graph of a function of x. Explain.
A graph of y = f(x) follows. No formula for f is given. Graph the given equation.
14) y = f(2x)
15) y = -2f(x + 1) - 3
16) y = 2f(x)
17) Write the formula for the piecewise function whose graph is given below.
18) Function f is given below. Find f(-4) .
3x  2

f ( x)   x
2 x  1

f o r x  2
for
2 x 3
for
x3
19) Determine whether the function f(x) = x3 +3x2
is even, odd, or neither.
20) Factor completely x2 + 5x + 6
21) Find the power function that the graph of f(x) = - x2(x+2)2(x2-1) resembles for large values of |x|
22) Graph the polynomial function f(x) = 6x-x3-x5
23) Let f(x) = 7 , and g(x) = 4 . Find (fog)(x)
x8
5x
24) For the polynomial function, f ( x)  1 x 4 ( x 2  3)( x  6) , list each real zero and its multiplicity.
3
Determine whether the graph crosses or touches the x-axis at each x -intercept. Sketch the graph of f
25) Factor completely 15z2 - 8z – 16
26) Find the inverse of the given function. Justify why the inverse exists.
a) f(x) = 3x2 -6, x  0
2x  1
b) f ( x) 
3  5x
27) Analyze the graph of the function f(x) = -2x3 (x - 3)4 (x + 2)2 as follows
a) Determine the end behavior: Find the power function that the graph of f resembles for large values of x.
b)Find the x-intercepts
c) Determine whether the graph crosses or touches the x –axis at each x-intercept
d) Use the information in a)-c) to sketch the graph of f.
28) How can the graph of f(x) = -(x-2)2 + 3 be obtained from the graph of y = x2? List the transformations in
the correct order.
29) Use the Remainder Theorem to find f(-1/2) for f(x) = 2x4 – 5x3- x2 + 3x + 2
30) Let f ( x) 1  4 x . Determine i) the domain of the function, ii) the range of the function, iii) the
domain of the inverse and iv) the range of the inverse.
31) Find functions f and g so that h(x) = (f o g)(x) where h(x) =
1
2x  1
32) The graph of the function f is shown below. Graph, using transformations, g(x) = f(-x) + 3
33) Determine the domain and range of the function.
34) For the function f(x) = 3 , construct and simplify the difference quotient f ( x  h)  f ( x) .
x 5
h
In problems 35-36, state whether the function is a polynomial function or not. If it is, give its degree. If it is
not, tell why not.
x4 1
35) f(x) =
x2
36) f(x) = -5x5- 2x4 +4x2+ 2
37) Analyze the graph of the function f(x) = -x2 (x - 1)(x + 3) as follows
a) Determine the end behavior: Find the power function that the graph of f resembles for large values of x.
b)Find the x-intercepts
c) Determine whether the graph crosses or touches the x –axis at each x-intercept
d) Use the information in a)-c) to sketch the graph of f.
38) Given
f ( x) 
2 Find the domain of the composite function f og.
2 and
g ( x) 
3x  1
x
39) Use synthetic division to determine whether x + 4 is a factor of the polynomial 3x3- 2x2 - 10x + 14
40) Factor completely 4x2 – 81
41)Graph f(x) = (x+3)3 -4 to determine whether f is a one-to-one function. If it is, find its inverse.
42) Determine whether the graph is the graph of an even function, an odd function, or a function that is
neither even nor odd.
43) Graph the function.
3x  2 f o r

f ( x)   x
for
2 x  1 f o r

x  2
2 x3
x3
44) Given the graph of a function f. Find the intervals of x on which f(x) < 0
In problems 45-46, find the domain of the function.
45) f(x) = 16  x
46) f(x) =
1
x  3x  18
2
47) Given functions f(x) = 2 x and g(x) =
x 1
4 , determine the domain of f + g.
x2
48) Find the x- and y-intercepts of f(x) = -x2(x+5) (x2+3)
49) Divide using synthetic division (x5 + x3 -5)  (x-2)
50) Determine whether the function ,whose graph is given below, is one-to one. Explain
a)
b)
51) Begin by graphing the basic function f(x) =|x|. Then use transformations to graph
h(x) = - |x+5|
52) Begin by graphing the basic function f(x) =
x . Then use transformations to graph g(x) =
 x5
53) Factor completely an expression that occurs in calculus
2(x + 6)(x-5)3 + 6 (x+6)2 (x-5)2
54)The graph of a function f is given below. Determine whether this function has the inverse. If yes, draw the
graph of f -1.
55) Find two functions f and g (none identity function) so that ( f  g )( x)  h( x) if
a) h( x)  5 3x  8
b) h( x) | x 2  3 |
Answers
1) (x +7)(x + 2)(x-2)
2) 4 x  2  26 x
3x 2  5
3) 4x2 + 6x + 7
4) 8x2 - 6x + 2
5) (-2, 0)
6) (-7, -5) (-3, 0)
7) a) behaves like y = -3x5; rises to the left and falls to the right;
b) behaves like y = - x8 ; falls to the left and falls to the right.
8) (4x - 1)(16x2 + 4x + 1)
9) (5x + 3)(25x2- 15x + 9)
10) (a) For large values of |x|, the graph of f(x) will resemble the graph of y = 2x5.The graph will fall on the
left and rise on the right
(b) x-intercepts: (-1, 0) , (2, 0)
(c) The graph of f crosses the x-axis at (2, 0) and touches the x-axis at (-1, 0).
(d)
11)
12) function
13) not a function
14)
15)
16)
17)
2 x  6,  5  x  2

f ( x)   1
 2 x  3,  2  x  4
18) -10
19) Neither
20) (x + 2)(x + 3)
21) y = -x6
22)
23)
35 x
4  40 x
24) 0, multiplicity 4, touches x-axis;
-6, multiplicity 1, crosses x-axis;
3 , multiplicity 1, crosses x-axis;
- 3 , multiplicity 1, crosses x-axis
25) (3z - 4)(5z + 4)
26) a) f 1 ( x)  x  6 ; b) f ( x)  3x  1
3
2  5x
27) (a) For large values of |x|, the graph of f(x) will resemble the graph of y = -2x9. The graph rises on the left
and falls on the right
(b) x-intercepts: (-2, 0) , (0, 0), and (3, 0)
(c) The graph of f crosses the x-axis at (0, 0) and touches the x-axis at (-2, 0) and (3,0).
(d)
28) Shift it horizontally 2 units to the right. Reflect it across the x-axis. Shift it 3 units up.
29) 1
30) domain of f = (-∞, ¼]= range of f -1 ; range of f = [0,+∞) = domain of f -1
31) f(x) = 1 , g(x) = 2x -1
x
32)
33) domain: (-∞, 6]; range: [-9, +)
3
34)
( x  h  5)( x  5)
35) No; it is a ratio of polynomials
36) Yes; degree 5
37) (a) For large values of |x|, the graph of f(x) will resemble the graph of y = -x4.Graph will fall on the left
and right
(b) x-intercepts: (-3, 0) , (0, 0), and (1, 0)
(c) The graph of f crosses the x-axis at (1, 0) and (-3, 0) and touches the x-axis at (0, 0).
(d)
38) {x | x  1 / 3}
39) No
40) (2x + 9)(2x - 9)
1
3
41) Yes, f ( x)  x  4  3
42) Even
43)
44) (-,-3)  (-1,0)  (2, +)
45) {x|x ≤ 16} =(-∞, 16]
46) {x|x ≠ - 6 and x ≠ 3} = (-∞, - 6) ∪ (- 6, 3) ∪ (3, ∞)
47) (-∞, -2)  (-2, 1)  (1, ∞)
48) x-intercepts: - 5, 0; y-intercept: 0
49) x4+ 2x3 + 5x2 + 10x + 20 + 35
x2
50) a) no; b) yes
51)
52)
53) 2(x + 6)(x-5)2 (4x + 13)
54) yes,
55) a) f ( x)  5 x ; g ( x)  3x  8 ; b) f ( x) | x | ; g ( x)  x 2  3