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Name: ______________________________ Period ______ Sec5-3 #9-12all, 24-30 evens, 41,45,47,51 Inverse Functions (notes about these problems) Sec5-4 #2-16 evens, 30 & 32 Solve exponential and log equations (review problems, see page 352, Example 1 & page 353, Example 2) Review Box: Inverse Functions An inverse functions switches the role of the input and the output Use domain & range to seect the part of the of a function. dashed graph that is the inverse function. To find the equation for the inverse, switch the x and y values and Domain of the function: x 3 then solve for y. Range of the function: y 4 Function: Switch x & y: f ( x) x 3 4 The domain of the inverse function is the same as the range of the function. x y 3 4 Domain of the inverse: x 4 Solve for y: Add 4 x 4 y 3 Square both sides x 4 Add 3 f 1 ( x) x 4 3 2 The range of the inverse function is the same as the domain of the function. y 3 Domain of the inverse: y3 2 The inverse function is a reflection of the function over the line y x Problems from Text: (a) Find the inverse function, (b) graph the function and its inverse on the same coordinate axis, (c) describe the relationship between the graphs, (d) state the domain & range of 24. f ( x) 3x 26. f ( x) x 3 1 f and f 1 . Use your own paper. 28. f ( x) x 2 , x 0 30. f ( x) x 2 4, x 2 Existence of an Inverse Function PreCalculus: Not every function has an inverse function. In precalculus we used the horizontal line test to check that the function has exactly one x-value for every y-value (and therefore an inverse relationship that is a function). We said that functions that pass the horizontal line test are one-to-one so their inverse relationship would also be a function. Calculus: If a function is one-to-one then that function is either always increasing or always decreasing. Functions that are always increasing or always decreasing on their domain are called strictly monotonic. Strictly monotonic functions have inverse relationships that are functions. Use the first derivative to test if a function is strictly monotonic: always increasing or always decreasing and therefore has an inverse functions. Examples: Use the derivative to determine whether the function is strictly monotonic on its entire domain and therefore has an inverse function. Take the derivative: f ( x) 2 x x3 domain is all real numbers f ( x) 2 3x 2 This function is monotonic because the derivative is always negative. This functions inverse relationship would be an inverse function. Take the derivative: f ( x) ln( x 5) 1 f ( x) x5 domain x greater than 5 This function is monotonic because the derivative is always positive on its domain (x greater than 3). This functions inverse relationship would be an inverse function. Problems from Text: Use the derivative to determine whether the function is strictly monotonic on its entire domain and therefore has an inverse function 41. f ( x) 2 x x 3 45. f ( x) ln( x 3)