439 4.1 Inverse Functions –2 0 2 1 6 x y y = x2 + 2 This graph does not pass the horizontal line test. Figure 6 Finding the Equation of the Inverse of y =f 1x2 For a one-to-one function ƒ defined by an equation y = ƒ1x2, find the defining equation of the inverse as follows. (If necessary, replace ƒ1x2 with y first. Any restrictions on x and y should be considered.) Step 1 Interchange x and y. Step 2 Solve for y. Step 3 Replace y with ƒ-11x2. EXAMPLE 5 Finding Equations of Inverses Determine whether each equation defines a one-to-one function. If so, find the equation of the inverse. (a) ƒ1x2 = 2x + 5 (b) y = x2 + 2 (c) ƒ1x2 = 1x - 223 SOLUTION (a) The graph of y = 2x + 5 is a nonhorizontal line, so by the horizontal line test, ƒ is a one-to-one function. Find the equation of the inverse as follows. ƒ1x2 = 2x + 5 Given function y = 2x + 5 Let y = ƒ1x2. Step 1 x = 2y + 5 Interchange x and y. Step 2 x - 5 = 2y Subtract 5. y = x - 5 2 Divide by 2. Rewrite. Step 3 ƒ -11x2 = 1 2 x - 5 2 Replace y with ƒ -11x2. a - b c = A 1 cB a - b c Thus, the equation ƒ -11x2 = x - 5 2 = 1 2 x - 5 2 represents a linear function. In the function y = 2x + 5, the value of y is found by starting with a value of x, multiplying by 2, and adding 5. The equation ƒ -11x2 = x - 5 2 for the inverse subtracts 5 and then divides by 2. An inverse is used to “undo” what a function does to the variable x. (b) The equation y = x2 + 2 has a parabola opening up as its graph, so some horizontal lines will intersect the graph at two points. For example, both x = 1 and x = -1 correspond to y = 3. See Figure 6. Because of the presence of the x2-term, there are many pairs of x-values that correspond to the same y-value. This means that the function defined by y = x2 + 2 is not one-to-one and does not have an inverse. Proceeding with the steps for finding the equation of an inverse leads to y = x2 + 2 x = y2 + 2 Interchange x and y. x - 2 = y2 Solve for y. {2x - 2 = y. Square root property The last equation shows that there are two y-values for each choice of x greater than 2, indicating that this is not a function. (+1)+1* Solve for y. Remember both roots.
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