Math 135 Functions: The Inverse Solutions
1. The constant function: f(x) = c, c ∈ R. Domain (−∞, ∞) Range {c} Intervals of Increase none Intervals of Decrease none Turning Points none Local Maxima c, x = c Local Minima c, x = c Global Maxima c, x = c Global Minima c, x = c Symmetry y-axis
y f −1(x)
f(x) = c (−2, c) (1, c) (c, c)
(c, 1)
x
(c, −2)
Recall that the reflection about the line y = x of a horizontal line is a vertical line. Observe that the function f(x) = c is not one-to-one and hence f −1(x) is not a func- tion.
University of Hawai‘i at Manoa¯ 103 R Spring - 2014 Math 135 Functions: The Inverse Solutions
2. The linear function: f(x) = ax + b; a, b ∈ R. Domain (−∞, ∞) {b}, a = 0; Range (−∞, ∞), a 6= 0. none, a ≤ 0; Intervals of Increase (−∞, ∞), a > 0. none, a ≥ 0; Intervals of Decrease (−∞, ∞), a < 0. Turning Points none none, a 6= 0; Local Maxima b, a = 0. none, a 6= 0; Local Minima b, a = 0. none, a 6= 0; Global Maxima b, a = 0. none, a 6= 0; Global Minima b, a = 0. none, a 6= 0; Symmetry y-axis, a = 0.
y
(0, 2)
(0, 2 ) (x, x) 3 (2, 0) x 2 ( 3 , 0) −1 1 2 f (x) = − 3 x + 3
f(x) = −3x + 2
University of Hawai‘i at Manoa¯ 104 R Spring - 2014 Math 135 Functions: The Inverse Solutions
3. The square function: f(x) = x2. Domain (−∞, ∞) Range [0, ∞) Intervals of Increase [0, ∞) Intervals of Decrease (−∞, 0] Turning Points x = 0 Local Maxima none Local Minima x = 0 Global Maxima none Global Minima x = 0 Symmetry y-axis
f(x) = x2 y (−2, 4) (2, 4)
f −1(x)
(1, 1) (−1, 1)
x (0, 0)
(1, −1)
(4, −2)
Observe that f(x) = x2 is not one-to-one on its domain, hence f −1(x) is not a func- tion.
University of Hawai‘i at Manoa¯ 105 R Spring - 2014 Math 135 Functions: The Inverse Solutions
4. The cube function: f(x) = x3. Domain (−∞, ∞) Range (−∞, ∞) Intervals of Increase (−∞, ∞) Intervals of Decrease none Turning Points none Local Maxima none Local Minima none Global Maxima none Global Minima none Symmetry origin
f(x) = x3 y
√ f −1(x) = 3 x (1, 1)
(0, 0) x
(−1, −1)
University of Hawai‘i at Manoa¯ 106 R Spring - 2014 Math 135 Functions: The Inverse Solutions
1 5. The inverse function: f(x) = x . Domain (−∞, 0) ∪ (0, ∞) Range (−∞, 0) ∪ (0, ∞) Intervals of Increase none Intervals of Decrease (−∞, 0) ∪ (0, ∞) Turning Points none Local Maxima none Local Minima none Global Maxima none Global Minima none Symmetry origin
y x = 0
(1, 1) f(x) = 1 x x y = 0
(−1, −1)
1 −1 1 The function f(x) = x is its own inverse and f (x) = x .
University of Hawai‘i at Manoa¯ 107 R Spring - 2014 Math 135 Functions: The Inverse Solutions
1 6. The inverse square function: f(x) = x2 . Domain (−∞, 0) ∪ (0, ∞) Range (0, ∞) Intervals of Increase (−∞, 0) Intervals of Decrease (0, ∞) Turning Points none Local Maxima none Local Minima none Global Maxima none Global Minima none Symmetry y-axis
y x = 0
(−1, 1) (1, 1) f −1(x) 1 x f(x) = x2 y = 0
(1, −1)
1 −1 Observe that f(x) = x2 is not one-to-one on its domain, hence f (x) is not a func- tion.
University of Hawai‘i at Manoa¯ 108 R Spring - 2014 Math 135 Functions: The Inverse Solutions
√ 7. The square root function: f(x) = x. Domain [0, ∞) Range [0, ∞) Intervals of Increase [0, ∞) Intervals of Decrease none Turning Points none Local Maxima none Local Minima x = 0 Global Maxima none Global Minima x = 0 Symmetry none
f −1(x) = x2 y (2, 4)
(4, 2) √ f(x) = x
(1, 1)
(0, 0) x
Note how the domain affects the graph of the inverse function and compare this to the above problem with f(x) = x2.
University of Hawai‘i at Manoa¯ 109 R Spring - 2014 Math 135 Functions: The Inverse Solutions
√ 8. The cube root function: f(x) = 3 x. Domain (−∞, ∞) Range (−∞, ∞) Intervals of Increase (−∞, ∞) Intervals of Decrease none Turning Points none Local Maxima none Local Minima none Global Maxima none Global Minima none Symmetry origin
f −1(x) = x3 y
√ f(x) = 3 x (1, 1)
(0, 0) x
(−1, −1)
University of Hawai‘i at Manoa¯ 110 R Spring - 2014 Math 135 Functions: The Inverse Solutions
9. The absolute value function: f(x) = |x|. Domain (−∞, ∞) Range [0, ∞) Intervals of Increase [0, ∞) Intervals of Decrease (−∞, 0] Turning Points x = 0 Local Maxima none Local Minima x = 0 Global Maxima none Global Minima x = 0 Symmetry y-axis
y y = |x|
(1, 1) (−1, 1)
x (0, 0)
f −1(x)
Note that f(x) = |x| is not one-to-one and so f −1(x) is not a function.
University of Hawai‘i at Manoa¯ 111 R Spring - 2014