# 005 Sample Final A, Question 1

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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 ${\displaystyle f(x)=\tan ^{-1}(x)}$ is all real numbers.
f. (True/False) The function ${\displaystyle \log _{a}(x)}$ is defined for all real numbers.

a) False. Nothing in the definition of a geometric sequence requires the common ratio to be always positive. For example, ${\displaystyle a_{n}=(-a)^{n}}$
c) False. ${\displaystyle y=x^{2}}$ does not have an inverse.
d) True. ${\displaystyle cos^{2}(x)-cos(x)=0}$ has multiple solutions.
e) True. The domain of ${\displaystyle \tan ^{-1}(x)}$ is the range of ${\displaystyle \tan(x)}$
f) False. The domain of ${\displaystyle \log _{a}(x)}$ is the range of ${\displaystyle e^{x}}$