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NAME DATE 2-1 PERIOD Study Guide and Intervention Power and Radical Functions Power Functions A power function is any function of the form f(x) = ax where a and n are nonzero constant real numbers. For example, 1 − f(x) = 2x2, f(x) = x 2 , or f(x) = √x are power functions. n 1 3 Graph and analyze f(x) = − x . Describe the domain, 3 range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. Evaluate the function for several x-values in its domain. Then use a smooth curve to connect each of these points to complete the graph. x -3 -2 -1 0 1 2 3 8 f(x) -9 8 -− 1 -− 0 1 − 8 − 9 4 3 3 3 3 −8 Domain: (-∞, ∞) Range: (-∞, ∞) Intercept: 0 End behavior: lim f(x) = -∞ and lim f(x) = ∞ x → -∞ y 0 −4 Lesson 2-1 Example 4 8x −4 −8 x→∞ Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Continuity: continuous for all real numbers Increasing: (-∞, ∞) Exercises Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 1 -3 2. f(x) = − x 1. f(x) = 3x4 5 y 8 y 4 0 x −8 −4 0 4 8x −4 −8 Chapter 2 5 Glencoe Precalculus NAME DATE 2-1 Study Guide and Intervention PERIOD (continued) Power and Radical Functions Radical Functions and Equations A radical function is a function that has at least one radical expression containing the independent variable. 4 For example, f(x) = 2 √ x3 is a radical function. A radical equation is an equation in which the variable is in the radicand. To solve a radical equation, isolate the radical expression. Then raise each side of the equation to the power of the index of the radical. Then check for extraneous solutions. These are solutions that do not satisfy the original equation. 3 Solve 3 = √ x2 - 2x + 1 - 1. Example Step 1 3 3 = √ x2 - 2x + 1 - 1 3 4 = √ x - 2x + 1 2 64 = x2 - 2x + 1 2 Original equation Isolate the radical. Cube each side. 0 = x - 2x - 63 Subtract 64 from each side. 0 = (x - 9)(x + 7) Factor. x - 9 = 0 or x + 7 = 0 x=9 x = -7 Solve. Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. Step 2 Check both solutions. 3 3 = √ x2 - 2x + 1 - 1 Zero Product Property 3 2 3 = √x - 2x + 1 - 1 3 2 - 2(9) + 1 - 1 3 √(9) 3 2 3 √(-7) - 2(-7) + 1 - 1 3 64 - 1 3 √ 3 3 √ 64 - 1 34-1 34-1 3=3 3=3 Both solutions check, so the solutions are −7 and 9. Exercises Solve each equation. 3 1. √ x2 - 1 - 6 = -4 2. √ 6n - 3 = √ -15 + 7n 56 - x 3. 4x = 21 + √ 5 4. √ 40 - 4x + 15 = 17 Chapter 2 6 Glencoe Precalculus