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Math 143: College Algebra Name:
Sample Final Exam Instructor:
Fall 2007 Section:
You are allowed to use a couple of 3 ร 5 cards of notes. Calculators are allowed.
Part I consists of 10 multiple choice questions worth 5 points each. Record your answers by placing an X through one letter for each problem on this answer sheet.
Part II consists of 10 partial credit problems worth a total of 100 points. Show all your work on the page on which the question appears.
Do not remove this answer page โ you need to return the whole exam. You are allowed two hours to do the exam. You may leave earlier if you are finished.
Answer Sheet
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For Grading Use Only
1โ10 11 12 13 14 15
16 17 18 19 20 Total
Part I: Multiple Choice Questions
1 (5 points). Find the domain of the function y =
x^2 โ 1
(a) (โโ, โ1] โช [1, โ) (b) [โ 1 , 1] (c) (โโ, โ1) โช (1, โ) (d) (โ 1 , 1) (e) (1, โ)
2 (5 points). Determine which graph represents a one-to-one function.
x
y
x
y
x
y
(a) (b) (c)
x
y
x
y
(d) (e)
3 (5 points). Find the inverse function of f (x) =
2 x โ 1 x โ 3
(a) f โ^1 (x) =
x โ 3 2 x โ 1
(b) f โ^1 (x) =
1 โ 2 x x โ 3
(c) f โ^1 (x) =
2 x + 1 x + 3
(d) f โ^1 (x) =
3 x โ 1 x โ 2
(e) f โ^1 (x) =
x โ 2 3 x โ 1
7 (5 points). Find the vertical asymptote to the graph of the function y =
2 x โ 6 4 x + 8
(a) x = โ 2 (b) x = 12 (c) x = 2 (d) x = โ^34 (e) x = 3
8 (5 points). Find the solution set of the inequality ex^2 โ^3 โค 1.
(a) x โค โ 2 or x โฅ 2 (b) โ 2 โค x โค 2 (c) x โค โ
3 or x โฅ
(d) โ
3 โค x โค
(e) x โค 2
9 (5 points). Find the solution set of the inequality log 2 (x + 1) < 2.
(a) x < 1 (b) x > โ 1 (c) x < 3 (d) โ 1 < x < 3 (e) โ 1 โค x < 3
10 (5 points). The complex number i^25 is equal to
(a) 5 i (b) 1 (c) i (d) โ 1 (e) โi
Part II: Partial Credit Problems
11 (10 points). Questions (a) through (d) refer to the graph of the function y = f (x):
x
y
1
1
y = f (x)
(a) Write the domain of f using the interval notation [a, b]:.
(b) Write the range of f using the interval notation [a, b]:.
(c) Find the average rate of change of f over its domain.
Solution.
(d) On which interval is f decreasing?
Answer.
12 (10 points). Let f (x) = x^2 + 3x and g(x) = x โ 2.
(a) Find f (โ2).
Solution.
(b) Find the difference quotient
f (x + h) โ f (x) h
. Simplify the answer.
Solution.
(c) Find (f โฆ g)(x). Simplify the answer.
Solution.
15 (14 points). Questions (a) through (d) refer to the function:
y = ln(x + 1) โ 1
(a) Graph the function y = ln(x + 1) โ 1.
x
y
1
1
(b) Write the domain and range of y = ln(x + 1) โ 1 using the interval notation [a, b].
Domain: Range:
(c) Find an equation of the vertical asymptote to the graph of y = ln(x + 1) โ 1.
Vertical Asymptote:
(d) Find the x- and y-intercepts of the graph of y = ln(x + 1) โ 1. Show all your work.
Solution.
x-Intercept: y-Intercept:
16 (9 points). Suppose that b is a positive constant greater than 1, and let A, B, and C be defined as follows: logb 2 = A logb 3 = B logb 5 = C
Use the properties of logarithms to evaluate each expression in terms of A, B, and/or C.
(a) logb 12
Solution.
(b) logb 0. 8
Solution.
(c) log 3 b 3
Solution.
17 (8 points). Solve the equation log 5 (x โ 1) + log 5 (x + 3) = 1 for x.
Solution.
