440 CHAPTER 4 Inverse, Exponential, and Logarithmic Functions This graph passes the horizontal line test. f(x) = (x – 2)3 y 2 0 –8 8 1 x Figure 7 (c) Figure 7 shows that the horizontal line test assures us that this horizontal translation of the graph of the cubing function is one-to-one. ƒ1x2 = 1x - 223 Given function y = 1x - 223 Replace ƒ1x2 with y. Step 1 x = 1y - 223 Interchange x and y. Step 2 23 x = 23 1y - 223 Take the cube root on each side. 2 3 x = y - 2 23 a3 = a 2 3 x + 2 = y Add 2. Step 3 ƒ -11x = 23 x + 2 Replace y with ƒ -11x2. Rewrite. S Now Try Exercises 59(a), 63(a), and 65(a). Solve for y. (++1)++1* In the final line, we give the condition x ≠2 to avoid dividing by 0. (Note that 2 is not in the range of ƒ, so it is not in the domain of ƒ -1.) Step 3 ƒ -11x2 = 4x + 3 x - 2 , x ≠2 Replace y with ƒ -11x2. S Now Try Exercise 71(a). Solve for y. EXAMPLE 6 Finding the Equation of the Inverse of a Rational Function The following rational function is one-to-one. Find its inverse. ƒ1x2 = 2x + 3 x - 4 , x ≠4 SOLUTION ƒ1x2 = 2x + 3 x - 4 , x ≠4 Given function y = 2x + 3 x - 4 Replace ƒ1x2 with y. Step 1 x = 2y + 3 y - 4 , y ≠4 Interchange x and y. Step 2 x1y - 42 = 2y + 3 Multiply by y - 4. xy - 4x = 2y + 3 Distributive property xy - 2y = 4x + 3 Add 4x and -2y. y1x - 22 = 4x + 3 Factor out y. y = 4x + 3 x - 2 , x ≠2 Divide by x - 2. Pay close attention here. Graphing the Inverse of a Function f Whose Equation Is Known Step 1 Find some ordered pairs that are on the graph of ƒ. Step 2 Interchange x and y to find ordered pairs that are on the graph of ƒ -1. Step 3 Plot those points, and sketch the graph of ƒ -1 through them. (''''')+'''+*
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