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5-2: One-to-One Functions; Inverse Functions If f(x) = y is a function, its inverse f –1(x) is f(y) = x, where y becomes the domain and x becomes the range. If for each y in the domain of the inverse function there is a unique x in the range, it is a one-to-one function. If a horizontal line intersects the graph of a function f no more than once, then f is one-to-one. Only one-to-one functions have inverses. We can verify that f and f –1 are inverses showing that f ( f –1 (x)) = f -1 ( f (x)) = x. The graph of a function f and its inverse f –1 are symmetric with respect to the line y = x. To find the inverse of a function y = f(x), interchange x and y to obtain x = f(y) and solve for y in terms of x. Find the inverse and determine whether the inverse is a function: 1. [(Bob, 68), (Dave, 92), (Carol, 87), (Elaine, 74), (Chuck, 87)]. 2. [(-2, 5), (-1, 3), (3, 7), (4, 12) Use the horizontal line test to see whether f is one-to-one; if yes, graph its inverse: (3) (4) (5) y = f(x) = -2x + 3 10 8 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 -2 -4 -6 -8 -10 Verify that the functions f and g are inverses of each other: 1 6. f(x) = 3 – 2x; g(x) = ( x 3) 7. f(x) = 2x – 7; g(x) = x 7 2 2 8. f(x) = x2 5 3x 5 , g x 3 The following functions are one-to-one. Find the inverse and state the domain and range of each. 9. 2 x 3 x #9 Domain Range f ( x) 10. f(x) f -1(x) f x x 2 4, x 0 #10 Domain Range f(x) f -1(x) TRY THESE Which of the following graphs are one-to-one? Draw the graph of the inverse function f-1 for those that are one-to-one. 1. 2. 3. 4. Verify that the functions f and g are inverses of each other. 5. f x x 2 2 , x 2; g x x 2 6. f x x 5 ; g x 3 x 5 2x 3 1 2x Function f is one-to-one. Find its inverse; check your answer. State the domain of f and find its range using f-1. 2 7. f x x 3 , x 0 3 x2 6-2: Exponential Functions Exponential function is a function in the form f(x) = ax where a is a positive real number and a 1. The domain of f is the set of all real numbers. The laws of exponents for real (irrational) exponents are the same as those for integer and rational exponents. Irrational exponents are 1I truncated to a finite number of digits, so that ar ax. The number e is the number that F 1 J G n H nK approaches as n. f(x) = ax Range x-intercept y-intercept a>1 [0, ) None (0, 1) 0<a<1 [0, ) None (0, 1) Horizontal asymptote x-axis as x x-axis as x Characteristic Increasing, one-to-one Decreasing one-to-one Passes through (0, 1) (1, a) (0, 1) (1, a) Approximate to three decimal places: 1. 3 5 2. 2e Graph the following exponential functions; state the domain, range, and any horizontal asymptote: x 3. y = 2x 4. y = F 5. y = ex G1 IJ = 2-x H2 K 2x+2; y= y = 2x – 2 9. y = -ex; y = 2 - ex y = 2-x-2; y = 2-x + 2 Atmospheric pressure p measured in mm of mercury is related to the number of km h above sea level by the formula p = 760e-0.145h. Find the pressure at a height of 2 km and at a height of 10 km. Exponential Equations: If au = av, then u = v. 1 2 x 10. 5 1 5 11. 1 2 1 x 4 12. e4 x e x e12 2