Review For Test 1 MAT146
1. For the real valued functions f(x) = 1  x and g(x) = x + 4, find the composition f ◦ g and
specify its domain using interval notation.
2. The one-to-one function f is defined by f ( x) 
 4x  9
. Find f-1. Find the domain and range
7  3x
of f-1 using interval notation.
3.Below is the graph of a polynomial function f with real coefficients. Use the graph to answer
the following questions about f. All local extrema of f are shown on the graph.
a)
b)
c)
d)
The function f is increasing over which intervals?
The function f ahs local min and local max values at which x-values?
What is the sign of the leading coefficient of f?
What is a possible degree of f?
4. Perform the following division: (12x4 + 19x3 + 13x2 – 3x – 20) ÷ (3x +4).
5. Use synthetic division to find the quotient Q(x) and remainder R when
P(x) = − x3 + 3x2 + 6x – 8 is divided by x – 4.
6. The function below has at least one rational zero. Use this fact to find all zeros of the
function. f(x) = 2x3 + 5x2 – 6x – 9
7) Graph the rational function f ( x) 
 x2  5x  5
. Find x and y –intercepts, vertical and/or
x2
horizontal asymptotes.
8) Use the rational zeros theorem to list all possible rational zeros of the following.
g(x) = − 2x3 – 3x2 – 3x – 5
9) Find the lowest degree polynomial f(x) that has the indicated zeros:
3
−5 (multiplicity 3) and
4
10) For the polynomial below, −3 is a zero. g(x) = x3 – x2 – 11x + 3. Express g(x) as a product
of linear factors.
11) Find all other zeros of P(x) = x3 – 8x2 + 24x – 32, given that 2 + 2i is a zero.
12) State the end behavior of the following polynomials:
a) P(x) = x3 – 5x2 + 2x + 6
b) P(x) = x4 + x3 – 3x2 + 5x – 7
c) P(x) = −x4 – 4x3 + 8x2 – 6x +2
d) P(x) = −x3 + 3x2 – 5x + 10
13) Determine whether x – 1 is a factor of 3x4 – 2x3 + 5x – 6
5
using synthetic division.
2
15) Show that x = 5 is a zero of P(x) = x3 – 5x2 + 4x – 20 and then find all other zeros.
14) Evaluate P(x) = 4x3 – 12x2 – 7x + 10 for x =
16) Use the upper and lower bound theorem to decide if the given positive number is an upper
bound and the given negative number is a lower bound for the real zeros of P(x).
P(x) = x4 – 4x3 + 8x2 + 2x + 1; x = 5 and x = −2
17) Use the location theorem to prove that the polynomial P(x) = x4 – x3 – 9x2 + 9x + 4 has a
zero in the interval (2, 3).
18) Solve the given polynomial inequality: a) 3x2 ≤ 39
b) 7x2 > 8x – x3
19) Look at problems 21 – 26 on page 333 in your book.
20) Find a polynomial P(x) of lowest degree, with leading coefficient 1, that has the indicated
set of zeros. Leave in factored form. Indicate the degree of the polynomial.
a) (2 – 3i), (2 + 3i), −4(multiplicity 3)
b) i 3 (multiplicity 2),  i 3 (multiplicity 2), 5 (multiplicity 5)
21) Write P(x) = x3 – x2 + 25x – 25 as a product of linear factors.
22) Find all zeros exactly for the following polynomial equation: 2x3 – 10x2 + 12x – 4 = 0
23) Solve the rational inequality :
24) Graph P(x) =
3x  7
0
x2  9
3x
; find x and y intercepts, vertical and horizontal asymptotes.
4x  4