Exercise 1.4.19 (19-20)

Question

Find the exponential function $f(x)=C b^{x}$ whose graph is given.

Khagan Gulusoy · Accepted Answer

\begin{enumerate}
  \setcounter{enumi}{18}
  \item Since the type of the desired function is 
$$\forall x \in \mathbb{R}: \quad f(x)=C b^{x}$$

for some positive real $C$ and $b\in \mathbb{R}$ ($C$ must be positive since the outputs of this function are positive for any inputs). Using the coordinates of the points that belong to the graph, we can write the following system:

$$
\begin{gathered}
\left\{\begin{array} { l } 
{ C b ^ { 2 } = 6 } \
{ C b ^ { 3 } = 2 4 }
\end{array} \iff \left\{\begin{array} { l } 
{ C b = 6 } \
{ C b ^ { 3 } = 2 4 }
\end{array} \iff \left\{\begin{array}{l}
C b=6 \
C b^{3}: C b=24: 6
\end{array} \iffight.ight.ight. \
\left\{\begin{array} { l } 
{ C b = 6 } \
{ b ^ { 2 } = 4 }
\end{array} \iff \left\{\begin{array} { l } 
{ C b = 6 } \
{ b = 2 	ext { or } b = - 2 }
\end{array} \iff \left\{\begin{array}{lll}
C=3 & 	ext { or } C=-3 \
b=2 & 	ext { or } b=-2 .
\end{array}ight.ight.ight.
\end{gathered}
$$

Since $C$ must be positive, we have:

$$
C=3 ; \quad b=2 	ext {. }
$$

Thus, the desired function is:

$$\forall x \in \mathbb{R}:\quad f(x)=3 \cdot 2^{x}.$$
\end{enumerate}

\begin{enumerate}
  \setcounter{enumi}{19}
  \item Since the type of the desired function is

$$
\forall x \in \mathbb{R}:\quad f(x)=C b^{x}
$$

for some positive real $C$ and $b \in \mathbb{R}$ (C must be positive since the outputs of this function are positive for any inputs). Using the coordinates of the points that belong to the graph, we can write the following system:

$$
\begin{aligned}
\forall x \in \mathbb{R}:\left\{\begin{array} { l } 
{ C b ^ { - 1 } = 3 } \
{ C b ^ { 1 } = \frac { 4 } { 3 } }
\end{array} \Leftrightarrow \left\{\begin{array} { l } 
{ C b ^ { - 1 } \cdot C b ^ { 1 } = 3 \cdot \frac { 4 } { 3 } } \
{ C b = \frac { 4 } { 3 } }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
C^{2}=4 \
C b=\frac{4}{3}
\end{array} \Leftrightarrowight.ight.ight. \
\left\{\begin{array} { l } 
{ C = 2 	ext { or } C = - 2 } \
{ C b = 4 / 3 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
C=2 	ext { or } C=-2 \
b=2 / 3 	ext { or } b=-2 / 3 .
\end{array}ight.ight.
\end{aligned}
$$

Since $C$ must be positive, we have:

$$	ext{$C=2$ and $b= \frac{2}{3}$.}$$

Thus, the desired function is:

$$\forall x \in \mathbb{R}: f(x)=2 \cdot\left(\frac{2}{3}ight)^{x}.$$
\end{enumerate}

Exercise 1.4.19 (19-20)

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Exercise 1.4.19 (19-20)

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