Chapter 7.6, Problem 68HP

To determine

To examine the given cases

Explanation of Solution

Given:

Parent Function: $f(x)=a{b}^x$

Let, a be positive for the analysis .

Calculation:

Case1: $g(x)=a{b}^{-x}$where $b>1$

Let $f(x)=2{\left(3\right)}^x$

Hence, $g(x)=2{\left(3\right)}^{-x}$

Calculation for graph:

Consider $f(x)=2{\left(3\right)}^x$

Values of x	Values of f (x)
0	2
1	6
2	18
-1	0.67
3	54

From the above table the graph can be potted.

Calculation for graph:

Consider $f(x)=2{\left(3\right)}^{-x}$

Values of x	Values of f (x)
0	2
-1	6
-2	18
1	0.67
-3	54

From the above table the graph can be potted.

Plotting both the graph on the same plane:

  

Interpretation:

From the above graph it is clear that $g(x)$ will be exact reflection of $f(x)$ about y -axis.

Case2: $g(x)=a{b}^{-x}$where $0<b<1$

Let $f(x)=2{\left(\frac{1}{3}\right)}^x$

Hence, $g(x)=2{\left(\frac{1}{3}\right)}^{-x}$

Calculation for graph:

Consider $f(x)=2{\left(\frac{1}{3}\right)}^x$

Values of x	Values of f (x)
0	2
-1	6
-2	18
1	0.67
-3	54

From the above table the graph can be potted.

Calculation for graph:

Consider $g(x)=2{\left(\frac{1}{3}\right)}^{-x}$

Values of x	Values of f (x)
0	2
1	6
2	18
-1	0.67
3	54

From the above table the graph can be potted.

Plotting both the graph on the same plane:

  

Interpretation:

From the above graph it is clear that $g(x)$ will be exact reflection of $f(x)$ about y -axis.