Chapter 5: Inverse, Exponential, and Logarithmic Functions
5.1:
In
v
erse Functions
1.
Different
x
-values always produce different
y
-values, therefore yes, it is one-to-one.
2.
Different
x
-values always produce different
y
-values, therefore yes, it is one-to-one.
3.
Choosing 2 and
as values for
x
yields:
.
Since different values of
x
produce the same value for
the function is not one-to-one.
4.
Choosing 2 and
as values for
x
yields:
.
Since different values
of
x
produce the same value for
the function is not one-to-one.
5.
Choosing 6 and
as values for
x
yields:
.
Since different values of
x
produce the same value for
the function is not one-to-one.
6.
Choosing 10 and
as values for
x
yields:
.
Since different values of
x
produce the same value for
the function is not one-to-one.
7.
Every horizontal line will intersect the graph at exactly one point, therefore yes, it is one-to-one.
8.
Every horizontal line will intersect the graph at exactly one point, therefore yes, it is one-to-one.
9.
There are horizontal lines that will intersect the graph at more than one point, therefore it is not one-to-one.
10. A certain horizontal line intersects the whole horizontal graph (more than one point), therefore it is not one-to-one.
11. Every horizontal line will intersect the graph at exactly one point, therefore yes, it is one-to-one.
12. Every horizontal line will intersect the graph at exactly one point, therefore yes, it is one-to-one.
13. Every horizontal line will intersect the graph at exactly one point, therefore yes, it is one-to-one.
14. Every horizontal line will intersect the graph at exactly one point, therefore yes, it is one-to-one.
15. A certain horizontal line intersects the horizontal graph when
(more than 1 point), therefore it is not one-to-one.
16. Since different
x
-values greater than zero all produce the same
, it is not one-to-one.
17. Choosing 4 and 0 as values for
x
yields:
.
Since different values
of
x
produce the same value for
the function is not one-to-one.
18. Choosing 0 and
as values for
x
yields:
.
Since different values of
x
produce the same value for
the function is not one-to-one.
19. Different
x
-values always produce different
y
-values, therefore yes, it is one-to-one.
20. Different
x
-values always produce different
y
-values, therefore yes, it is one-to-one.
21. Different
x
-values always produce different
y
-values, therefore yes, it is one-to-one.
22. Different
x
-values always produce different
y
-values, therefore yes, it is one-to-one.
23. Different
x
-values always produce different
y
-values, therefore yes, it is one-to-one.
24. All
x
-values produce the same
, therefore it is not one-to-one.
25. The graph fails the horizontal line test because the end behavior is either
or
.
f
1
x
2
5]
7
f
1
x
2
,
f
1
0
2
1
0
1
3
2
2
2
8
17 and
f
1
]
6
2
1
]
6
1
3
2
2
2
8
17
]
6
f
1
x
2
,
f
1
4
2
5
1
4
2
2
2
2
5
4 and
f
1
0
2
5
1
0
2
2
2
2
5
4
f
1
x
2
5
3
x
7
0
f
1
x
2
,
f
1
6
2
5
2
100
2
10
2
5
0 and
f
1
]
10
2
5
2
100
2
1
]
10
2
2
5
0
]
10
f
1
x
2
,
f
1
6
2
5
2
36
2
6
2
5
0 and
f
1
]
6
2
5
2
36
2
1
]
6
2
2
5
0
]
6
f
1
x
2
,
f
1
2
2
2
2
4 and
f
1
]
2
2
1
]
2
2
2
4
]
2
f
1
x
2
,
f
1
2
2
5
2
2
5
4 and
f
1
]
2
2
5
1
]
2
2
2
5
4
]
2