2.
5 Inverse Functions
51
Use the horizontal line test to determine whether the function whose graph is shown is one-to-one. 1.
2.
Determine whether the function is one-to-one using the denition of one-to-one. Example f (x) = 2x + 3 Solution Let f (a) = f (b). Show that a = b. f (a) 2a + 3 2a a Therefore, f is one-to-one. = = = = f (b) 2b + 3 2b b
52 3. h(x) = 4x 9 4. f (x) = 1 x + 1 2 5. 6. x+2 x+3 x1 x+2
2 FUNCTIONS AND GRAPHS
Find the inverse function using the switch and solve method. 7. f (x) = 2x + 5 8. f (x) = 5x + 2 9. f (x) = 3x 1 10. f (x) = 11. f (x) = 2x 1 x6 x+2 x3
Find f g(x) and g f (x). Then determine whether f and g are inverse functions of each other. 1 1 + 3, g(x) = x x3
3
12. f (x) = 13. f (x) =
x2 , g(x) = 5x3 + 2 5
2.5 Inverse Functions The graph of f is given. Sketch the graph of f 1 .
53
14.
15.
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2 FUNCTIONS AND GRAPHS
2.5 Inverse Functions
55
Use the horizontal line test to determine whether the function whose graph is shown is one-to-one. 1.
The function is not one-to-one because it fails the horizontal line test. 2.
The function is one-to-one because it passes the horizontal line test. Determine whether the function is one-to-one using the denition of one-to-one. Example f (x) = 2x + 3 Solution Let f (a) = f (b). Show that a = b. f (a) 2a + 3 2a a = = = = f (b) 2b + 3 2b b
56 Therefore, f is one-to-one. 3. h(x) = 4x 9
2 FUNCTIONS AND GRAPHS
Solution Let f (a) = f (b). Show that a = b. f (a) 4a 9 4a a Therefore, f is one-to-one. 4. f (x) = 1 x + 1 2 Solution Let f (a) = f (b). Show that a = b. f (a) = f (b) 1 1 a+1 = b+1 2 2 1 1 a = b 2 2 a = b Therefore, f is one-to-one. 5. x+2 x+3 Solution Let f (a) = f (b). Show that a = b. f (a) a+2 a+3 (a + 2)(b + 3) ab + 3a + 2b + 6 3a + 2b 3a 2a a = f (b) b+2 = b+3 = (b + 2)(a + 3) = ab + 3b + 2a + 6 = 3b + 2a = 3b 2b = b = = = = f (b) 4b 9 4b b
2.5 Inverse Functions Therefore, f is one-to-one. 6. x1 x+2 Solution Let f (a) = f (b). Show that a = b. f (a) a1 a+2 (a 1)(b + 2) ab + 2a b 2 2a b 2a + a 3a a = f (b) b1 = b+2 = (b 1)(a + 2) = ab + 2b a 2 = 2b a = 2b + b = 3b = b
57
Therefore, f is one-to-one. Find the inverse function using the switch and solve method. 7. f (x) = 2x + 5 Solution
y = 2x + 5 x = 2y + 5 < Switch x 5 = 2y < Solve 5 1 y = x+ 2 2 5 1 1 f (x) = x + 2 2 8. f (x) = 5x + 2 Solution
58
2 FUNCTIONS AND GRAPHS
y = 5x + 2 x = 5y + 2 x 2 = 5y 1 2 y = x 5 5 1 2 f 1 (x) = x 5 5 3x 1
9. f (x) =
Solution
y = x =
3x 1, 3y 1,
y0 x0
x2 = 3y 1, x 0 3y = x2 + 1, x 0 x2 + 1 y = , x0 3 x2 + 1 , x0 f 1 (x) = 3 2x 1 x6 Solution 2x 1 x6 2y 1 x = y6 x(y 6) = 2y 1 xy 6x = 2y 1 y =
10. f (x) =
2.5 Inverse Functions xy 2y = 6x 1 y(x 2) = 6x 1 6x 1 y = x2 6x 1 f 1 (x) = x2
59
11. f (x) =
x+2 x3
Solution
y = x = x(y 3) xy 3x xy y y(x 1) = = = =
y = f 1 (x) =
x+2 x3 y+2 y3 y+2 y+2 3x + 2 3x + 2 3x + 2 x1 3x + 2 x1
Find f g(x) and g f (x). Then determine whether f and g are inverse functions of each other.
12. f (x) =
1 1 + 3, g(x) = x x3
Solution
60
2 FUNCTIONS AND GRAPHS
f (g(x)) = f = 1
1 x3 +3
1 x3
= (x 3) + 3 = x Yes, the functions are inverses. x2 , g(x) = 5x3 + 2 5
g(f (x)) = g (1/x + 3) 1 = (1/x + 3) 3 1 =x = 1/x
13. f (x) =
Solution
3 x 2 g(f (x)) = g 5
x 2 +2 = 5 5
3
= 5
(x 2) + 2 = (x 2) + 2 = x 5
Yes, the functions are inverses.
2.5 Inverse Functions The graph of f is given. Sketch the graph of f 1 .
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14. Solution
15. Solution
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2 FUNCTIONS AND GRAPHS
