Answers to Section 26. One-to-One Functions and Their Inverses

7. (a) the graph is a complete parabola facing up, so it does not pass the horizontal line test; (b)
fx is one-to-one on 3, ; (d) gx 3 1 2x 10 ; domain 5, ; range: 3, . The
2
graph of fx restricted to 3, is in red, and the graph of gx is in blue:
12
10
8
6
4
2
0
2 4 6 8 10 12
x
8. (a) the graph is a complete parabola facing up, so it does not pass the horizontal line test; (b)
fx is one-to-one on , 8/3; (d) gx 8 3 1 6x 14 ; domain 7/3, ; range:
3
, 8/3. The graph of fx restricted to , 8/3 is in red, and the graph of gx is in
blue:
10
8
6
4
2
-2
0
2 4 x 6 8 10
-2
9. (a) f / x 3x 2 5 5 0, so fx is monotonic increasing for all reals, hence 1-1. (b) By
trial and error, g 9 1 and g 15 2. (c) g / 9 1
f / 1 1 and
8
g / 15 1
f / 2 17 1 .
10. (a) f / x 3x 2 12x 15 3x 2 4x 4 4 15 3x 2 2 3 3 0, so fx is
monotonic increasing for all reals, hence 1-1. (b) By trial and error, g 17 2 and
g 19 1. (c) g / 17 1
f / 2 1 3 and g / 19 1
f / 1 30 1 .
11. (a) f / x 2 sinx 5 3 0, so fx is monotonic decreasing for all reals. (b) The 5x
term should give you a hint that g 5/2 /2 and g 15/2 3/2. (c)
g / 5/2 1
f / /2 3 1 and g / 15/2 1
f / 3/2 3 1 also.
12. (a) f / x 14 12 cos4x 2 0, so fx is monotonic increasing for all reals. (b) The
14x term should give you a hint that g 7 /2 and g 7/2 /4. (c)
g / 7 1
f / /2 1 2 and g / 7/2 1
f / /4 26 1 .
Section 26. One-to-One Functions and Their Inverses 84