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MATH 100-D200 Instructor: R. Pyke
Midterm 2, Version 2, November 6, 2008
Last Name:
First Name:
SFU Student email : @sfu.ca
1. DO NOT LIFT UP THE COVER PAGE UNTIL INSTRUCTED.
2. Clearly explain your answer. No credit will be given for just writing down
the answer.
3. If the answer space provided is not sufficient, write your answer on the back
of the previous page.
4. Ordinary Scientific Calculators ONLY are allowed.
NO GRAPHING CALCULATORS ALLOWED.
5. Copying someone else’s test, or deliberately exposing written
papers to the view of others is forbidden and will result in a
score of zero and disciplinary action.
Question Score Max 1 4 2 4 3 9 4 10 5 10 6 10 7 4 8 4 Total 55
(1) [Marks: 4] Express f (x) =
∣∣∣ ∣^23 xx^ −+ 2^4
∣∣∣ ∣ as a piecewise function.
(3) [Marks: 9] Below are the graphs of two functions f (x) and g(x). The domain of f (x) is [1, ∞) and the range of f (x) is [− 1 , ∞). The domain of g(x) is (−∞, ∞) and the range of g(x) is [0, ∞).
f(x)
g(x)
(a) Using these graphs, determine (approximately) the value of ( f ◦ g) (0) and ( g ◦ f ) (4). Explain your reasoning by referring to the graphs.
(b) Determine an (approximate) x for which ( f ◦ g) (x) = − 1 /2. Explain your reasoning by referring to the graphs.
(b) Determine the domain and range of f ◦ g.
(4) [Marks: 10] (a) Prove that h(x) = (^1) −|^3 x 2 |x 2 is not one-to-one.
(b) Prove that f (x) = 3 + √x^3 − 7 is one-to-one.
(c) Find f −^1 (x).
(d) Verify that ( f −^1 ◦ f ) (x) = x.
(6) [Marks: 9] Sketch the graph of R(x) = −^4 x^23 −x + 2^8 x^ + 12. Find all intercepts, asymptotes, and determine how the graph approaches the asymptotes.
(7) [Marks: 4] Given that 2 is a root of p(x) = x^4 − 5 x^3 + 7x^2 + 3x − 10, find the complete factorization of p(x).
