Instructions. (14 points) Use proper mathematical
notation, sound writing, and good organization
to provide solutions to each of the following exercises in the space provided.
Calculators are not allowed on this examination.
(1pt) 1. Find all real solutions, if any, of the equation
Solution: The least common denominator (LCD) is
(x-5)(x-6), so first clear fractions
by multiplying both sides of the equation by the LCD to obtain the equation
Then simplify this quadratic equation to standard form:
Finally, solve this quadratic equation by using the
quadratic formula to obtain the
solutions 
Since each of these values does
not cause division by
zero in the original equation, they are both valid solutions.
(1pt) 2. Find all real solutions, if any, of the equation
Solution:
Solving this quadratic equation yields x = -5,-6.
However, -5 does not solve the original equation.
(1pt) 3. Find all real solutions, if any, of the equation
Solution: Let 
.
Substitute these in
to obtain
Factor.
Thus,
Substitute back 
.
Raise each equation to the third power.
Thus,
Both answers check.
(1pt) 4. Solve the inequality:
. Sketch your answer on a number line , the
describe
your answer in interval notation.
Solution:
In interval notation, the solution is
(1pt) 5. What is the domain of the function
? Ex press your answer in interval
notation.
Solution: The square root of a negative number is not
defined as a real number . Thus,
92x - 27 must be greater than or equal to zero.
implies that
, so
the domain is the interval 
.
(1pt) 6. Given the function f(x) = x^2 -6x+1, find the
average rate of change of f from 3 to x ,
and simplify your answer.
Solution: The average rate of change from 3 to x is the
difference quotient
Factor and cancel.
Hence,
provided x ≠ 3.
(1pt) 7. Given the graph of y = f(x) below, sketch the
graph of 
Solution: No solution provided
(1pt) 8. Find the range of the function f(x) = x^2 + 8x
-
7. Express your answer in interval
notation.
Solution: The graph opens upward since a = 1 > 0, and the
vertex is at (h, k), where
and 
.
Thus, the range is 
.
(1pt) 9. Given
Sketch the graph of f(x). Shade the solution of f(x) > 0
on the horizontal axis, then
describe your solution using interval notation.
Solution: f(x) is a polynomial of even degree with leading
coefficient 1. Therefore, the
graph is above the x-axis on the interval 
.
The graph then crosses the x-axis at
zeros of odd multiplicity (x = 4 and x = 13 in this case), and just touches the
x-axis at
zeros of even multiplicity (x = 8). Therefore, the graph is above the x-axis on
the set
(1pt) 10. Given
sketch the graph of f. Label all asymptotes with their
equations and any critical points
with their coordinates.
Solution: Vertical asymptotes occur where the simplified function is not
defined.
so there are no vertical asymptotes.
(1pt) 11. Given that x = -3 is a root of
, list all roots of the polynomial p(x).
Solution: Given that x = -3 is a root of
, we use synthetic division to find:
Thus, p(x) will factor as follows:
Use the quadratic formula to find the roots of the
quadratic factor.
(1pt) 12. Consider the function
Sketch the graph of f. Label all asymptotes with their
equations and any critical points
with their coordinates.
Solution: Solution here.
(1pt) 13. Calculate 
.
Express your answer in the form a + bi.
Solution: To calculate the quotient of two complex
numbers, multiply the numerator
and denominator by the complex conjugate of the denominator.
(1pt) 14. Given that x = -3 is a root of
, list all roots (real or complex)
of the polynomial p(x).
Solution: Given that x = -3 is a root of
, we use synthetic
division to find:
Thus, p(x) will factor as follows:
Use the quadratic formula to find the roots of the
quadratic factor.
