Download Review for test 1 (MWF) 1) Factor completely x3+ 7x2 – 4x– 28 2

Review for test 1 (MWF)
1) Factor completely x3+ 7x2 – 4x – 28
2) 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.
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) Find the power function that the graph of f(x) = - x2(x+2)2(x2-1) resembles for large values of |x|.
8) Factor completely 64 x3- 1
9) Factor completely
125 x3+ 27
10) 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.
11) Determine whether or not the graph below is the graph of a function of x. Explain.
12) Determine whether or not the graph below is the graph of a function of x. Explain.
A graph of y = f(x) follows. No formula for f is given. Graph the given equation.
13) y = f(2x)
14) y = -2f(x + 1) - 3
15) y = 2f(x)
16) Write the formula for the piecewise function whose graph is given below.
17) 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
18) Determine whether the function f(x) = x3 +3x2
is even, odd, or neither.
19) Factor completely x2 + 5x + 6
1 4 2
x ( x  3)( x  6) , list each real zero and its multiplicity. Determine whether the
3
graph crosses or touches the x-axis at each x -intercept.
21) Let f(x) = 7 , and g(x) = 4 . Find (fog)(x)
20) For the polynomial function, f ( x) 
x8
5x
22) Find the x- and y-intercepts of f(x) = -x2(x+5) (x2+3)
23) Factor completely 15z2 - 8z - 16
24) Graph f(x) = (x+3)3 -4 to determine whether f is a one-to-one function. If it is, find its inverse.
25) 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.
26) Find the inverse of the given function. Justify why the inverse exists.
a) f(x) = 3x2 -6, x  0
b) f ( x)  2 x  1
3  5x
27) Find functions f and g so that h(x) = (f og)(x) where h(x) =
1
2x  1
28) The graph of the function f is shown below. Graph, using transformations, g(x) = f(-x) + 3
29) Determine the domain and range of the function.
30) For the function f(x) = 3 , construct and simplify the difference quotient f ( x  h)  f ( x) .
x 5
h
In problems 31-32, 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
31) f(x) =
x2
32) f(x) = -5x5- 2x4 +4x2+ 2
33) Given
f ( x) 
2 Find the domain of the composite function f og.
2 and
g ( x) 
3x  1
x
34) 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.
35) Factor completely 4x2 - 81
36) Determine whether the graph is the graph of an even function, an odd function, or a function that is neither even nor odd.
Justify your answer.
37) 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
38) Given the graph of a function f. Find the intervals of x on which f(x) < 0
In problems 39-40, find the domain of the function.
39) f(x) = 16  x
40) f(x) =
1
x 2  3x  18
41) Given functions f(x) = 2 x and g(x) =
x 1
4 , determine the domain of f + g.
x2
42) Determine whether the function ,whose graph is given below, is one-to one. Explain
a)
b)
43) Begin by graphing the basic function f(x) =|x|. Then use transformations to graph
h(x) = - |x+5|
44) Begin by graphing the basic function f(x) =
x . Then use transformations to graph g(x) =
 x5
45) Factor completely an expression that occurs in calculus
2(x + 6)(x-5)3 + 6 (x+6)2 (x-5)2
46) The graph of a function f is given below. Determine whether this function has the inverse. If yes, draw the graph of f -1.
47) 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) (a) For large values of |x|, the graph of f(x) will resemble the graph of y = -x4.
(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)
3) 4x2 + 6x + 7
4) 8x2 - 6x + 2
5) (-2, 0)
6) (-7, -5) (-3, 0)
7) y = -x6
8) (4x - 1)(16x2 + 4x + 1)
9) (5x + 3)(25x2- 15x + 9)
10)
11) function
12) not a function
13)
14)
15)
16)
2 x  6,  5  x  2

f ( x)   1
 2 x  3,  2  x  4
17) -10
18) Neither
19) (x + 2)(x + 3)
20) 0, multiplicity 4, touches x-axis;
-6, multiplicity 1, crosses x-axis;
3 , multiplicity 1, crosses x-axis;
- 3 , multiplicity 1, crosses x-axis
21)
35 x
4  40 x
22) x-intercepts: - 5, 0; y-intercept: 0
23) (3z - 4)(5z + 4)
1
3
24) Yes, f ( x)  x  4  3
25) Shift it horizontally 2 units to the right. Reflect it across the x-axis. Shift it 3 units up.
26) a) f
1
( x) 
x6
3x  1
; b) f ( x) 
3
2  5x
27) f(x) = 1 , g(x) = 2x -1
x
28)
29) domain: (-∞, 6]; range: [-9, +)
3
30)
( x  h  5)( x  5)
31) No; it is a ratio of polynomials
32) Yes; degree 5
33) {x | x  1 / 3}
34) domain of f = (-∞, ¼]= range of f -1 ; range of f = [0,+∞) = domain of f -1
35) (2x + 9)(2x - 9)
36) Even
37)
38) (-,-3)  (-1,0)  (2, +)
39) {x|x ≤ 16} =(-∞, 16]
40) {x|x ≠ - 6 and x ≠ 3} = (-∞, - 6) ∪ (- 6, 3) ∪ (3, ∞)
41) (-∞, -2)  (-2, 1)  (1, ∞)
42) a) no; b) yes
43)
44)
45) 2(x + 6)(x-5)2 (4x + 13)
46) yes,
47 a) f ( x)  5 x ; g ( x)  3x  8 ; b) f ( x) | x | ; g ( x)  x 2  3