Review for test 1
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) Find a rational zero of the polynomial function f(x) = x3 + 2 x2 - 6x – 4 and use it to find all the zeros of the function. Factor the
polynomial function.
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 is a graph of a function of x. Explain.
12) 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.
13) y = f(2x)
14) y = -2f(x + 1) - 3
15) y = 2f(x)
16) Write an equation for the piecewise function.
17) Function f is given below. Find f(-4) .
3x  2 fo r x  2

f ( x)   x
fo r  2  x  3 ; f(-4)
2 x  1 fo r x  3

18) Determine whether the function f(x) = x3 +3x2
is even, odd, or neither.
19) Factor completely x2 + 5x + 6
20) Using synthetic division, determine whether the numbers 7, 3 are zeros of the polynomial f(x) = x3 – 11x2 + 36x - 36.
21) Let f(x) = 7 , and
x8
g(x) = 4 . Find (f∘g)(x)
5x
22) Solve the equation in the real number system.
2x4 – 2x3 + x2 - 5x - 10 = 0
23) Factor completely 15z2 - 8z - 16
24) List all possible rational zeros for the function f(x) = 6x4 + 5x3 – 7x2 + 2
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 only the rational zeros of f(x) = x4 + 2x3 + 2x2- 4x - 8
27) Find functions f and g so that h(x) = (f ∘ g)(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
State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not.
31) f(x) =
x4 1
x2
32) f(x) = -5x5- 2x4 +4x2+ 2
33) Given
f ( x) 
2
2 and
Find the domain of the composite function f∘g.
g ( x) 
3x  1
x
34) Use synthetic division to determine whether x + 4 is a factor of the polynomial 3x3- 2x2 - 10x + 14
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.
37) Graph the function.
3x  2

f ( x)   x
2 x  1

fo r x  2
fo r  2  x  3
fo r x  3
38) Given the graph of a function f. Find the intervals of x on which f(x) < 0
Find the domain of the function.
39) f(x) = 16  x
40) f(x) =
1
x  3x  18
2
41) Given functions f(x) = 2 x and g(x) = 4 , determine the domain of f + g.
x 1
x2
42) Divide using synthetic division (x5 + x3 -5)  (x-2)
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) =
45) Factor completely an expression that occurs in calculus
2(x + 6)(x-5)3 + 6 (x+6)2 (x-5)2
 x5
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) {2, -2 + 2 , -3 - 2 }; ( x  2)( x  2  2 )( x  2  2 )
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
 x  3,  2  x  4

 2
17) -10
18) Neither
19) (x + 2)(x + 3)
20) No; yes
21) 35 x
4  40 x
22) {-1, 2}
23) (3z - 4)(5z + 4)
24)  1 , 1 , 1 , 2 , ± 1, ± 2
6 3 2 3
25) Shift it horizontally 2 units to the right. Reflect it across the x-axis. Shift it 3 units up.
26) No rational zeros
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) No
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) x4+ 2x3 + 5x2 + 10x + 20 + 35
x2
43)
44)
45) 2(x + 6)(x-5)2 (4x + 13)