Chapter 3.1, Problem 73E

Solution

To find: The conditions that expression $f\left(a,b,c\right)$ is an exponential function.

For $a=2,b=1/3$ the function $f\left(a,b,c=x\right)=a.b^x$ is a decreasing exponential function.

Given data: The value of c is x and $f\left(a,b,c\right)=a\cdot b^c$ .

Method/Formula used:

An expression $f\left(x\right)=k\cdot m^x$

with k and m as the constants is called the exponential expression.

Calculation:

The given expression $f\left(a,b,c\right)$ is

  $f\left(a,b,c\right)=a\cdot b^c$ … (1)

Here, value of c is x , therefore, substitute x for c in result (1),

  $f\left(a,b,x\right)=a\cdot b^x$ … (2)

Now choose constants a and b such that $a=2$ and $b=\frac{1}{3}$ , so substitute 2 for a and 1/3 for b in the result (2),

  $\begin{array}{c}f\left(2,1/3,x\right)=2\cdot {\left(\frac{1}{3}\right)}^x\\ =2\cdot {\left(3^{-1}\right)}^x\\ =2\cdot 3^{-x}\end{array}$

  

Therefore, the function $f\left(2,1/3,x\right)=2\cdot 3^{-x}$ is the required decreasing exponential expression.

The graph of decreasing function is shown in Fig. 1