Question

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

20.

Short answer

The exponential equation is \(y = 2 \cdot {\left( {\frac{2}{3}} \right)^x}\).

Step-by-step solution

Obtain the value of constant 

From the graph, the points are \(\left( { - 1,3} \right)\), and \(\left( {1,\frac{4}{3}} \right)\).

Consider the equation\(y = f\left( x \right) = C{b^x}\).

Use the point\(\left( { - 1,3} \right)\)to get the value of constant as shown below:

\(\begin{aligned}3 &= C{b^{ - 1}}\\C &= 3b\end{aligned}\)

Thus, \(C = 3b\).

Obtain the value of constant

Consider the equation \(y = f\left( x \right) = C{b^x}\).

Use the point\(\left( {1,\frac{4}{3}} \right)\)to get the value of constant as shown below:

\(\begin{aligned}\frac{4}{3} &= C{b^1}\\Cb &= \frac{4}{3}\end{aligned}\)

Substitute\(3b\)for \(C\) in the equation \(Cb = \frac{4}{3}\)to get the value of\(b\)as shown below:

\(\begin{aligned}\left( {3b} \right) \cdot b &= \frac{4}{3}\\3{b^2} &= \frac{4}{3}\\{b^2} &= \frac{4}{9}\\b &= \pm \frac{2}{3}\end{aligned}\)

Since\(b > 0\), so\(b = \frac{2}{3}\).

The value of\(C\)is shown below:

\(\begin{aligned}C &= 3\left( {\frac{2}{3}} \right)\\ &= 2\end{aligned}\)

So, the exponential equation becomes \(y = 2 \cdot {\left( {\frac{2}{3}} \right)^x}\).

Thus, the exponential equation is \(y = 2 \cdot {\left( {\frac{2}{3}} \right)^x}\).