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

Answers to Section 26. One-to-One Functions and Their Inverses
1. gx 3 5 x ; domain: , ; range: , .
2. gx 1 x
2x
3. gx 4x x 1 3
; domain: , 0 0, ; range: , 2 2, .
; domain: , 1 1, ; range: , 4 4, .
4. gx 7x 2 3x 5 ; domain: , 2/3 2/3, ; range: , 7/3 7/3, .
5. gx x2 8 ; domain: 0, ; range: 8/3, .
3
6. (a) The function passes the horizontal line test:
4
3
2
1
-4 -2
0
x
2
-1
-2
(b) domain: , 7/2; range: , 3; (c) gx 1 3x 1 2 x2 ; domain: , 3; range:
, 7/2; (d) The graphs together:
4
-4 -2 x2
0
2
-2
-4
Section 26. One-to-One Functions and Their Inverses 83

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

13. (a) f / x 2 3 2 0, so fx is monotonic increasing on 0, . (b)
2 x
f 1 x 1 2 x 9 8 3 8x 9 . Finding this is a bit tricky. First solve for y with the
8
3 8x 9
quadratic formula, and we get y . To make sure that y is positive, we
4
pick the choice. Squaring both sides gives us f 1 x. (c) Since fx is monotonic increasing
on 0, and obviously increases without bound, and f0 0, the range of fx is also
0, . Thus, the domain of g x is 0, , and the range is also 0, .
(d) g / x 1 2 3
2 8x 9 , so g / 14
11 4 . (e) g 14 4.
(f) g / 14 1
f / 4 1
2 3
4 , as expected.
11
2 4
Section 26. One-to-One Functions and Their Inverses 85