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