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Math 135 Functions: the Inverse Solutions 1. the Constant Function: F Math 135 Functions: The Inverse Solutions 1. The constant function: f(x) = c; c 2 R. Domain (−∞; 1) Range fcg 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 2 R. Domain (−∞; 1) fbg, a = 0; Range (−∞; 1), a 6= 0. none, a ≤ 0; Intervals of Increase (−∞; 1), a > 0. none, a ≥ 0; Intervals of Decrease (−∞; 1), 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 (−∞; 1) Range [0; 1) Intervals of Increase [0; 1) 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 (−∞; 1) Range (−∞; 1) Intervals of Increase (−∞; 1) 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 p 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; 1) Range (−∞; 0) [ (0; 1) Intervals of Increase none Intervals of Decrease (−∞; 0) [ (0; 1) 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; 1) Range (0; 1) Intervals of Increase (−∞; 0) Intervals of Decrease (0; 1) 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 p 7. The square root function: f(x) = x. Domain [0; 1) Range [0; 1) Intervals of Increase [0; 1) 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) p 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 p 8. The cube root function: f(x) = 3 x. Domain (−∞; 1) Range (−∞; 1) Intervals of Increase (−∞; 1) 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 p 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) = jxj. Domain (−∞; 1) Range [0; 1) Intervals of Increase [0; 1) 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 = jxj (1; 1) (−1; 1) x (0; 0) f −1(x) Note that f(x) = jxj is not one-to-one and so f −1(x) is not a function. University of Hawai‘i at Manoa¯ 111 R Spring - 2014.
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