Chapter 7.1, Problem 97E

To determine

To sketch the graph of the system of equations: $y=b^x\text{and}y=x^b$ by taking $b=2\text{and}b=4$ draw the interpretation about the point of intersection of the graphs.

Answer to Problem 97E

For $b>1$ , the curves always intersect at atleast two points.

Explanation of Solution

Given information: The system of equations: $y=b^x\text{and}y=x^b$

Also, draw the graph at $b=2\text{and}b=4$

Calculation:

(a)  Using graphing utility, the graph of the system of equations are different values of b are:

At $b=2$ , the graph of the system of equations are:

  

At $b=4$ , the graph of the system of equations are:

  

From the graphs, it is concluded that at $b=2$ , the curves intersect at three points where at $b=4$ , the curves intersect at two points.

(b) From these graphs, the conjecture about the point of intersection is that at point of intersection, x-intercept is same in both type of curves and the final value of y-intercept is also same.

Hence, for $b>1$ , the cuves always intersect at atleast two points.