Exercise 1.4.17 (17-18)

Question

Find the domain of each function.

\begin{enumerate}
  \setcounter{enumi}{16}
  \item \begin{enumerate}[label=(\alph*)]
\item $f(x) =\frac{1-e^{x^{2}}}{1-e^{1-x^{2}}}$

\item  $f(x)=\frac{1+x}{e^{\cos x}}$
\end{enumerate}
\end{enumerate}

\begin{enumerate}
  \setcounter{enumi}{17}
  \item \begin{enumerate}[label=(\alph*)]
\item $g(t) =\sqrt{10^{t}-100}$

\item  $g(t)=\sin \left(e^{t}-1\right)$
\end{enumerate}
\end{enumerate}

Khagan Gulusoy · Accepted Answer

\begin{enumerate}
  \setcounter{enumi}{16}
  \item \begin{enumerate}[label=(\alph*)]
\item Since the denominator of a fraction cannot be equal to zero, the domain of the function $f$ is:

$$
\begin{aligned}
& \left\{x \mid 1-e^{1-x^{2}} 
eq 0ight\}=\left\{x \mid e^{1-x^{2}} 
eq 1ight\}=\left\{x \mid 1-x^{2} 
eq 0ight\}=\left\{x \mid x^{2} 
eq 1ight\}= \
& \quad=\{x \mid x 
eq-1 	ext { and } x 
eq 1\}=(-\infty,-1) \cup(-1,1) \cup(1,+\infty) .
\end{aligned}
$$

\item  Since the denominator of a fraction cannot be equal to zero and there are no restrictions for the domain of a natural exponential function, the domain of $f$ is:

$$
\left\{x \mid e^{\cos x} 
eq 0ight\}=\{x \mid x \in \mathbb{R}\}=\mathbb{R} 	ext {. }
$$
\end{enumerate}

\item \begin{enumerate}[label=(\alph*)]
\item Since the term under the root cannot be smaller than zero, the domain of $f$ is:

$$
\left\{t \mid 10^{t}-100 \geqslant 0ight\}=\left\{t \mid 10^{t} \geqslant 100ight\}=\left\{t \mid t \geqslant 2\}=[2,+\infty)ight. 	ext {. }
$$

\item  Since there are no restrictions for the domains of sine and natural exponential functions, the domain of the function $g$ is $\mathbb{R}$.
\end{enumerate}
\end{enumerate}

Exercise 1.4.17 (17-18)

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Exercise 1.4.17 (17-18)

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