2-1 Power and Radical Functions

Transcription

1 Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 15. h(x) = x 3 Evaluate the function for several x-values in its domain. x h(x) Use these points to construct a graph. Since the power is negative, the function will be undefined at x = 0, and D = (, 0) (0, ). All values of y are included on the graph except 0, so R = (, 0) (0, ). The only time the graph intersects the axes is when it goes through the origin, so the x- and y-intercepts are both 0. The y-values approach zero as x approaches negative or positive infinity, so. and The graph has an infinite discontinuity at x = 0. As you read the graph from left to right, it is going down from negative infinity to 0, and then going down again from 0 to positive infinity, so the graph is decreasing on (, 0) (0, ). Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing esolutions Manual - Powered by Cognero Page 1

2 or decreasing. 34. f (x) = 3 Evaluate the function for several x-values in its domain. x f(x) Use these points to construct a graph. x = 2. To find the y-intercept, substitute 0 for x in the original equation. The y-intercept is at y The y-value approaches positive infinity as x approaches positive infinity, so. There are no breaks, holes, or gaps in the graph, so it is continuous over the domain [ 2, ). As you read the graph from left to right, it is going up from 2 to positive infinity, so the graph is increasing on ( 2, ). Since it is an even-degree radical function, the domain is restricted to nonnegative values for the radicand, 6 + 3x. Solve for x when the radicand is 0 to find the restriction on the domain and the x- intercept. The domain is restricted to values of x greater than or equal to 2: D = [ 2, ). This is an increasing function, so use the restriction on the domain to find the minimum value for y. The graph contains all values of y from 0 to infinity, so R = [0, ). The first calculation shows that the x-intercept is at esolutions Manual - Powered by Cognero Page 2

3 43. AGRICULTURAL SCIENCE The net energy NE m required to maintain the body weight of beef cattle, in megacalories (Mcal) per day, is estimated by the formula where m is the animal s mass in kilograms. One megacalorie is equal to one million calories. a. Find the net energy per day required to maintain a 400-kilogram steer. b. If 0.96 megacalorie of energy is provided per pound of whole grain corn, how much corn does a 400-kilogram steer need to consume daily to maintain its body weight? a. Substitute m = 400. Solve each equation x = + 2 Since the each side of the equation was raised to a power, check the solutions in the original equation. x = 0 The net energy per day required to maintain a 400- kilogram steer is approximately 6.89 Mcal. b. Divide 6.89 Mcal by the 0.96 Mcal found in a pound of whole grain corn to find the total amount of corn necessary to maintain a 400-kilogram steer. x= 4 It will take about 7.18 pounds of corn to maintain a 400-kilogram steer. Neither value for x is a solution for the original equation. Thus, there is no solution. The graph below of the expressions on either side of the equation shows that there is no intersection, so the left side will never equal the right side. [-10, 10] scl: 1 by [-10, 10] scl: 1 esolutions Manual - Powered by Cognero Page 3

4 54. 1 = Without using a calculator, match each graph with the appropriate function. a. b. g(x) = x 6 c. h(x) = 4x 3 d. 70. The end behavior of the graph indicates that n is positive and even. Also, a is positive. The equation that matches this description is g(x) = x 6. Since the each side of the equation was raised to a power, check the solutions in the original equation. x = 1 The answer is b. 71. There is an infinite discontinuity at x = 0. This indicates that the power is negative. The equation that matches this description is h(x) = 4x 3. The answer is c. One solution checks and the other solution does not. Therefore, the solution is x = 1. esolutions Manual - Powered by Cognero Page 4

5 72. The domain of the graph is restricted to nonnegative values. This indicates that the function has a rational exponent with an even denominator or is a radical function with an even value for n. The equation that matches this description is. The answer is a. 73. The end behavior and the continuity of the graph indicate that it is a radical function with an odd value for n. The equation that matches this description is. The answer is d. esolutions Manual - Powered by Cognero Page 5