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Math 25 Calculus I Quiz 1
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                               Sept 6, 2005 \hfill  SHOW ALL
WORK



1.) Let $f(x) = {x \over x + 2}$.  Calculate the following.  Simplify your answer.

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i.) $f(x + 3) = \underline{~~{x +3 \over x + 5}~~}$


$f(x + 3) = {x +3 \over x + 3 + 2}= {x +3 \over x + 5}$.  

ii.)  $f^{-1}(x) = \underline{~~{2x \over 1-x}~~}$


$y = {x \over x + 2}$

Switch $x$ and $y$:  $x = {y \over y + 2}$

Solve for $y$:  $x(y + 2) = y$

$xy + 2x = y$

$2x = y - xy$

$2x = y(1 - x)$

$y = {2x \over 1-x}$

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iii.)  State the domain of $f$:   $\underline{ ~~{\cal R} - \{-2\}~~}$

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iv.)  State the range of $f$:  $\underline{~~{\cal R} - \{1\}~~}$ 

Note range of $f$ = domain of $f^{-1}$.  Hence your answer to iv.) will depend on your answer to ii.)  We will grade iv.) based on your answer to ii.)  

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v.)  Is $f$ one-to-one?  $\underline{~~Yes~~}$ 

Since the inverse exists, it must be one-to-one.
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2.) Match the following functions to their graphs:


a.)  $y = e^x - 1$ \hfil  b.) $y = ln(x - 1)$ \hfil c.)  $y = e^{-x} - 1$

d.)  $y = cos(x - {\pi \over 2})$ \hfil e.)  $y = sin(x + {\pi \over 2})$ 
\hfil f.) $y = 
\sqrt{x + 1}

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3.)  Circle T for true and F for false. 


 ~~~~~ F~~~ 6i.)  If $f$ is a function, the $f(s + t) = f(s) + f(t)$.

 ~~~~~ F~~~ 6ii.) If $f$ is a one-to-one function, the $f^{-1}(x) = {1 \over f(x)}$.

 ~~~~~ F~~~ 6iii.) If $a > 0$ and $b > 0$, $ln(a + b) = ln(a)ln(b)$

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