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) 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
6) Determine whether or not the graph is a graph of a function of x. Explain.
7) 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.
8) y = f(2x)
9) y = -2f(x + 1) - 3
10) y = 2f(x)
11) Write the formula for the piecewise function whose graph is given below.
12) 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
13) Determine whether the function f(x) = x3 +3x2
is even, odd, or neither.
14 ) Analyze the graph of the function f(x) = -x2 (x - 1)(x + 3) as follows:
(a)
(b)
(c)
(d)
Determine the end behavior: find the power function that the graph of f resembles for large values of |x|.
Find the x- intercepts of the graph.
Determine whether the graph crosses or touches the x-axis at each x-intercept.
Use the information obtained in (a) - (c) to sketch the graph of
g(x) = 4 . Find (f∘g)(x)
15) Let f(x) = 7 , and
x8
5x
1 4 2
x ( x  3)( x  6) , list each real zero and its multiplicity. Determine whether
3
the graph crosses or touches the x-axis at each x -intercept.
16) For the polynomial function, f ( x) 
17) 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.
18) Find the x- and y-intercepts of f(x) = -x2(x+5) (x2+3)
19) Find functions f and g so that h(x) = (f ∘ g)(x) where h(x) =
1
2x  1
20) The graph of the function f is shown below. Graph, using transformations, g(x) = f(-x) + 3
21) Determine the domain and range of the function.
22) For the function f(x) = 3 , construct and simplify the difference quotient f ( x  h)  f ( x) .
x 5
h
In problems 23-24, state whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why
not.
23) f(x) =
x4 1
x2
24) f(x) = -5x5- 2x4 +4x2+ 2
f ( x) 
25) Given
2
2 and
Find the domain of the composite function f∘g.
g ( x) 
3x  1
x
26) Determine whether the graph is the graph of an even function, an odd function, or a function that is neither even nor
odd.
27) Graph the function.
3x  2

f ( x)   x
2 x  1

f o r x  2
for  2  x  3
for
x3
28) Given the graph of a function f. Find the intervals of x on which f(x) < 0
In problems 29-30, find the domain of the function.
29) f(x) =
16  x
30) f(x) =
1
x  3x  18
2
31) Given functions f(x) = 2 x and g(x) =
x 1
4 , determine the domain of f + g.
x2
32) Determine the end behavior of the polynomial function
f(x) = - x2(x+2)2(x2-1)
33) Begin by graphing the basic function f(x) =|x|. Then use transformations to graph
34) Begin by graphing the basic function f(x) =
h(x) = - |x+5|
x . Then use transformations to graph g(x) =
 x5
Answers
1) 4x2 + 6x + 7
2) 8x2 - 6x + 2
3) (-2, 0)
4) (-7, -5) (-3, 0)
5)
6) function
7) not a function
8)
9)
10)
11)
2 x  6,  5  x  2

f ( x)   1
 x  3,  2  x  4

 2
12) -10
13) Neither
14) (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)
15)
35 x
4  40 x
16) 0, multiplicity 4, touches x-axis;
-6, multiplicity 1, crosses x-axis;
3 , multiplicity 1, crosses x-axis;
- 3 , multiplicity 1, crosses x-axis
17) Shift it horizontally 2 units to the right. Reflect it across the x-axis. Shift it 3 units up.
18) x-intercepts: - 5, 0; y-intercept: 0
19) f(x) = 1 , g(x) = 2x -1
x
20)
21) domain: (-∞, 6]; range: [-9, +)
22)
3
( x  h  5)( x  5)
23) No; it is a ratio of polynomials
24) Yes; degree 5
25) {x | x  1 / 3}
26) Even
27)
28) (-,-3)  (-1,0)  (2, +)
29) {x|x ≤ 16} =(-∞, 16]
30) {x|x ≠ - 6 and x ≠ 3} = (-∞, - 6) ∪ (- 6, 3) ∪ (3, ∞)
31) (-∞, -2)  (-2, 1)  (1, ∞)
32) For x with large |x|, the graph of f resembles the graph of y = -x6, that is the graph falls to the left and falls to the right
33)
34)