Exercise 1.3.45 (45-50)

Question

Express the function in the form $f=g$.\begin{enumerate}\setcounter{enumi}{44}
\item $F(x)=\left(2 x+x^2ight)^4$
\item $F(x)=\cos ^2 x$
\item $F(x)=\frac{\sqrt[3]{x}}{1+\sqrt[3]{x}}$
\item $G(x)=\sqrt[3]{\frac{x}{1+x}}$
\item $v(t)=\sec \left(t^2ight) 	an \left(t^2ight)$
\item $H(x)=\sqrt{1+\sqrt{x}}$
\end{enumerate}

Khagan Gulusoy · Accepted Answer

\begin{enumerate}\item[45.] To find the value of $F(x)$ for some real number $x$, we first find the value of $x^2+2 x$ and then elevate it to the power of 4. Therefore, we define the function $f$ and $g$ as follows:
$$
\begin{array}{ll}
& f(x)=x^4 \
& g(x)=x^2+2 x .
\end{array}
$$
\item[46.] To find the value of $\cos ^2 x$ for some particular real number $x$, we first find cos $x$ and then elevate it to the power of 2. Thus, we define the functions $f$ and $g$ as follows:
$$
\begin{array}{ll}
& f(x)=x^2 . \
& g(x)=\cos x .
\end{array}
$$
\item [47.] To find the value of $F(x)$ for some real number $x$, we first find $\sqrt[3]{x}$ and then the value of $\sqrt[3]{x}/(\sqrt[3]{x}+1)$. Therefore $g$ should be defined as follows:
$$
\begin{aligned}
&f(x)=x /(x+1) . \% \forall x \in (-\infty, -1)\cup (-1,+\infty): \quad 
& g(x)=\sqrt[3]{x} .
\end{aligned}
$$
\item[48.] To find the value of $G(x)$ for some real number $x$, we first find $\frac{x}{1+x}$ and then take the cubic root of this value. Therefore, the functions $f$ and $g$ should be defined as follows:
$$
\begin{aligned}
&  f(x)=\frac{x}{1+x} . \%\forall x \in (-\infty, -1)\cup (-1,+\infty): \quad
& g(x)=\sqrt[3]{x} .
\end{aligned}
$$
\item[49.] To find the value of $v(t)$ for some real number $t$ we first find the value of $x^2$ and then multiply the secant and the tangent of this value. Therefore, the functions $f$ and $g$ should be defined as follows:
$$
\begin{array}{ll}
& f(t)=\sec t \cdot 	an t . \
& g(t)=t^2 .
\end{array}
$$
\item[50.] To find the value of $H(x)$ for some real number $x$, we first find the value of $\sqrt{x}+1$ and then take the square root of the resulting value. Therefore, functions $f$ and $g$ must be defined as follows:
$$
\begin{aligned}
&f(x)=\sqrt{1+x} . \
&g(x)=\sqrt{x} .
\end{aligned}
$$
\end{enumerate}

Exercise 1.3.45 (45-50)

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Exercise 1.3.45 (45-50)

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