Practice Exam for Final
1. (Alvarenga, Kimberly Destiny) Determine whether the statement
is true or false.
(a) The domain of f (x) =
1
x2 +1
is the set of all real numbers.
(b) i100 = −1.
√
(c) The function f (x) = x3 1 − x2 is even.
(d) If f (x) = x + 1 and g(x) = 2x, then (f ◦ g)(x) = (g ◦ f )(x).
(e) If a square matrix has an entire row of zeros, the determinant
will always be zero.
(f)
(n+1)!
n!
= n + 1.
2. Simplify each expression.
(a) (Backes, Dana Elizabeth)
1
−
64
− 1
3
(b) (Backes, Dana Elizabeth)
√
x − 2√1 x
√
x
(c) (Block, Taylor Anna)
2 − x + 2x2 − x3
x2 − 4
(d) (Block, Taylor Anna)
2
3
−
1+i 1−i
3. Solve the equation.
(a) (Brown, Ryan Thomas)
|x2 − 3x| = −4x + 6
(b) (Chaudhary, Mohit)
4
6
15
+
=
x − 1 3x + 1
3x + 1
(c) (Cho, Hyun Soo)
e2x − 3ex − 4 = 0
(d) (Cordova, Whitney Danielle)
log(x + 4) − log x = log(x + 2)
4. (Dehaan, Drew Morgan) Solve the inequality and sketch the
solution on the real number line.
(a)
1 < −2x − 3 < 9
(b)
|3 − 4x| ≥ 9
5. (Fuller, Brett Scott) Write the standard form of the equation of
the circle with the given characteristics :
Endpoints of a diameter (0, 0), (6, 8).
6. (Goldstein, Elyse Gabrielle) (a) Find the equation of the line
passing through the points.
(−3, −2), (1, 6)
(b) Find the equation of the line through (2, 2) that is parallel to
the line found in (a).
7. (Hansgen, Jed Wells) Identify the parent function g, describe the
sequence of transformations from g to f , and sketch the graph of f .
(a)
f (x) = [x] + 3,
where [x] is the greatest integer less than or equal to x.
(b)
f (x) = − log5 (x − 1) + 2
8. (Jacobsen, Jeffrey C) Find and sketch f ◦ g and g ◦ f .
f (x) =
1
, g(x) = x + 3
x
9. (Kim, Tae Young) Find the inverse of the function f and sketch
the graph of the inverse function.
f (x) =
x+1
x−2
10. (Larson, Rachel Anne) Write the standard form of the quadratic
function whose graph has the vertex (3, 4) and passes through (1, 2).
11. Sketch the graph of the function by applying the Leading Coefficient Test, finding the zeros of the polynomial, and plotting sufficient
solution points.
(a) (Mansell, Karl W)
f (x) = x3 − 9x
(b) (Mcfarlane, Elle Babette)
f (x) = x4 − 4x2
12. (Pham, Andy Van) Divide (x3 − x2 − 14x + 11) by (x − 4).
(a) Use long division to divide.
(b) Use synthetic division to divide.
13. (Rabner, Jared Daniel) Show that (x − 5), (x + 4) are factors of
f (x) = x4 − 4x3 − 15x2 + 58x − 40. Then find the remaining factors of
f (x).
14. (Riemondy, Shane A) Find all the zeros of the function and
write the polynomial as a product of linear factors.
f (x) = x3 − 3x2 + 4x − 2
15. (Shelton, Hayden Karl) State the domain of the function, identify all intercepts, find any vertical and horizontal asymptotes, find
additional solution points, and sketch the graph.
x2 − 5x + 4
x2 − 4
16. (Tyler, Thomas Reuben) Write the partial fraction decomposition of the rational expression.
(a)
5
2
x +x−6
(b)
x2 − 1
x(x2 + 1)
17. (Walker, William Colby) Solve the system.
(a)
(
x − 2y = 0
3x − y 2 = 0
(b)
(
3x + 2y = 10
2x + 5y = 3
18. (Weber, Dore Lee) Find the augmented matrix and write in
row-echelon form or reduced row-echelon form to solve the system of
equations.



x
− 3z = −2
3x + y − 2z = 5

 2x + 2y + z = 4
19. (Wheeler, Elizabeth Bailey) Use the inverse matrix to solve the
system.
(
3x + 4y = −2
5x + 3y = 4
20. (Wilson, Kathryn Elizabeth) Find the determinant of the matrix


1 4 −2

0 
.
 3 2
−1 4
3
sol)
21. (Wolcott, David Aaron) Find a formula for an for the arithmetic
sequence
a6 = 6, a20 = 34,
and find the 20th partial sum of the sequence.
22. (Woolf, Ashley Kathleen) Find a formula for an for the geometric sequence
a1 = −2, a6 =
1
,
16
and find the 7th partial sum of the sequence.
23. Use Binomial theorem to expand (3x − 2y)4 .