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\textbf{Math 165: Honors Calculus I}

\textbf{Assignment 20}\quad \emph{Nov. 3, 1999}

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p. 149: 1--5 and

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6. Suppose $f$ and $g$ are functions such that $g(f(x)) = x$ for all $x$ in
the domain of $f$. Prove that $f$ is one-to-one.

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7. Let $f(x) = \int_1^x \frac{1}{t}\,dt$ for $x > 0$.
  \begin{enumerate}[a)]
  \item Prove that $f$ is strictly increasing and $f^{-1}(0) = 1$.

  \item Prove that $f(ab) = f(a) + f(b)$. (Hint: $\int_1^{ab}\frac{1}{t}\,dt
  =\int_1^{a}\frac{1}{t}\,dt + \int_a^{ab}\frac{1}{t}\,dt$, apply
  Theorem~1.19.)

  \item Prove that $f^{-1}(a+b) = f^{-1}(a)f^{-1}(b)$.
  \end{enumerate}

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8. Let $f$ be strictly decreasing on $[a,b]$. Prove that the inverse
$f^{-1}$  is strictly decreasing on $[f(b),f(a)]$.

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9. Let $f(x) = x^4 - 8 x^2 + 8$. Determine the intervals on which $f(x)$
is strictly increasing and strictly decreasing and find the inverse of
$f(x)$ on each of those intervals. Plot $f(x)$ and each of the inverses
as functions of $x$. (Hint: simplify $f(x)$ by `completing the square.')

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10. Determine the largest interval containing~$0$ on which the functions
$\sin(x)$ and $\tan(x)$ have inverses and plot the inverses on those
intervals. Explain why there is no interval containing~$0$ (i.e., no
interval of the form $[a,b]$ where $a<0$ and $b>0$) on which $\cos(x)$
has an inverse. Then find an interval on which $\cos(x)$ \emph{does}
have an inverse and plot the inverse there.

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