Chapter 1.5, Problem 21E

To determine

To find: The exponential function which is of the form $f\left(x\right)=C{b}^x$ for the given graph.

Answer to Problem 21E

The equation of the graph is $f\left(x\right)=3{\left(2\right)}^x$.

Explanation of Solution

It is given that the equation of the given graph is of the form $f\left(x\right)=C{b}^x$.

Here, the graph represents a curve which passes through the points (3, 24) and (1, 6).

Substitute these points in $f\left(x\right)=C{b}^x$,

$24=C{b}^3$ (1)

$6=C{b}^1$ (2)

Divide the equation (1) by equation (2) and obtain the value of b as follows.

$\begin{array}{l}\frac{24}{6}=\frac{C{b}^3}{C{b}^1}\\ 4=b^2\\ b=\pm 2\end{array}$

Ignore the negative value of b as it cannot take the negative values.

Thus, the value of b = 2.

Substitute $b=2$ in equation (1) and obtain the value of C.

$\begin{array}{l}24=C{2}^3\\ 24=8C\\ C=3\end{array}$

Therefore, the value of C = 3.

Substitute the value of b and C in $f\left(x\right)=C{b}^x$ and obtain the required equation.

Therefore, equation of the exponential function which passes through the points (3,24) and (1,6) is $f\left(x\right)=3{\left(2\right)}^x$.