Concept explainers
(a).
Make tables for f(x) and g(x) using n = 3 and n = 4.
(a).
Explanation of Solution
Given:
Calculation:
n =3 | n=4 | |||
x | ||||
-3 | -27 | -1.442 | 81 | No real solution |
-2 | -8 | -1.260 | 16 | No real solution |
-1 | -1 | -1 | 1 | No real solution |
0 | 0 | 0 | 0 | 0 |
1 | 1 | 1 | 1 | 1 |
2 | 8 | 1.260 | 16 | 1.189 |
3 | 27 | 1.442 | 81 | 1.316 |
Since the radicand of a radical with even index has to be non-negative.
(b).
Graph the equations.
(b).
Explanation of Solution
Calculation:
From part (a),
n =3 | n=4 | |||
x | ||||
-3 | -27 | -1.442 | 81 | No real solution |
-2 | -8 | -1.260 | 16 | No real solution |
-1 | -1 | -1 | 1 | No real solution |
0 | 0 | 0 | 0 | 0 |
1 | 1 | 1 | 1 | 1 |
2 | 8 | 1.260 | 16 | 1.189 |
3 | 27 | 1.442 | 81 | 1.316 |
Plot the points on the coordinate plane and graph the equations:
(c).
Find the equations that are functions and the functions that are one-to-one.
(c).
Answer to Problem 58PPS
All are functions and all except
Explanation of Solution
Given:
Calculation:
From part (b) ,
Since the graph of all the equations pass the vertical line test , that is any vertical line intersects the graph at only one point. So, all the equations are functions.
Since the graph of the function
(d).
Find the values of n for which f(x) and g(x) are inverses of each other.
(d).
Answer to Problem 58PPS
Explanation of Solution
Given:
Calculation:
From part (c) ,
Since
A function has an inverse if and only if it is one-to-one.
From the same rule, since
Also , we can observe from the graph :
If two functions are inverse of each other , their graph is symmetric along the line
Since the graphs of the functions
But , the graphs of the functions
(e).
Find the conclusions that you can make about
(e).
Answer to Problem 58PPS
For positive even values of n , the pair of functions are not inverses.
For positive odd values of n, the pair of functions are inverses.
Explanation of Solution
Given:
Calculation:
For positive even values of n , the pair of functions are not one-to −one . Hence are not inverses of each other. Here , the domain of
While for postitive odd values of n , the pair of functions are one-to-one .Hence they are inverses of each other.
Chapter 7 Solutions
Algebra 2
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