Question

Determine the exponential function which is of the form \(f(x) = C{b^x}\) for the given graph.

Short answer

The equation of the graph is \(f(x)=Cb^{x}\).

Step-by-step solution

Given data

The equation of the given graph is of the form \(f(x)=Cb^{x}\).

Concept of the domain of an exponential function

The domain is the set of all input values of the function for which the function is real and defined.

The domain of an exponential functions\(y = {a^x},\;a > 0\)and\(a \ne 1\)is the set of real values.

That is,\( - \infty < x < \infty \).

Find the value of \(C\)

Here, the graph represents a curve which passes through the points \((0,\;2)\) and \(\left( {2,\;\frac{2}{9}} \right)\).

Substitute \((0,2)\) in \(f(x) = C{b^x}\)as follows:

\(\begin{aligned}{c}2 &= C{b^0}\\C &= 2(1)\\{b^0} &= 1\quad \\C &= 2\end{aligned}\)

Find the value of \(b\)

Substitute \(\left( {2,\;\frac{2}{9}} \right)\) in \(f(x) = C{b^x}\).

\(\begin{aligned}{c}\frac{2}{9} &= C{b^2}\\\frac{2}{9} &= 2{b^2}\;\\C &= 2\end{aligned }\)

Simplify further by putting the value of \(C\)as shown below.

\(\begin{aligned }{c}{b^2} &= \frac{2}{{18}}\\{b^2} &= \frac{1}{9}\\b &= \pm \frac{1}{3}\end{aligned }\)

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

Thus, the value of \(b = \frac{1}{3}\).

Find the exponential function

Substitute the value of \(b = \frac{1}{3}\) and \(C = 2\) in \(f(x) = C{b^x}\) and obtain the required equation is \(f(x) = 2{\left( {\frac{1}{3}} \right)^x}.\)

Therefore, the equation of the exponential function which passes through the points \((0,\;2)\) and \(\left( {2,\;\frac{2}{9}} \right)\) is \(f(x) = 2{\left( {\frac{1}{3}} \right)^x}\).