005 Sample Final A, Question 1

Question Please circle either true or false,
    a. (True/False) In a geometric sequence, the common ratio is always positive.
    b. (True/False) A linear system of equations always has a solution.
    c. (True/False) Every function has an inverse.
    d. (True/False) Trigonometric equations do not always have unique solutions.
    e. (True/False) The domain of $f(x)=\tan ^{-1}(x)$ is all real numbers.
    f. (True/False) The function $\log _{a}(x)$ is defined for all real numbers.

Final Answers
a) False. Nothing in the definition of a geometric sequence requires the common ratio to be always positive. For example, $a_{n}=(-a)^{n}$
b) False. Linear systems only have a solution if the lines intersect. So y = x and y = x + 1 will never intersect because they are parallel.
c) False. $y=x^{2}$ does not have an inverse.
d) True. $cos^{2}(x)-cos(x)=0$ has multiple solutions.
e) True. The domain of $\tan ^{-1}(x)$ is the range of $\tan(x)$
f) False. The domain of $\log _{a}(x)$ is the range of $e^{x}$

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