Chapter 4: Rational, Power, and Root Functions 4.1 Rational Functions and Graphs 4.2 More on Rational Functions and Graphs 4.3 Rational Equations, Inequalities, Models, and Applications 4.4 Functions Defined by Powers and Roots 4.5 Equations, Inequalities, and Applications Involving Root Functions

4.5 Equations, Inequalities, and Applications Involving Root Functions Note: This property does not say that the two equations are equivalent. The new equation may have more solutions than the original. e.g. Power Property If P and Q are algebraic expressions, then every solution of the equation P = Q is also a solution of the equation Pn = Qn, for any positive integer n.

4.5 Solving Equations Involving Root Functions Solving an Equation Involving a Root Function Isolate a term involving a root on one side of the equation. Raise both sides of the equation to a power that will eliminate the radical or rational exponent. Solve the resulting equation. (If a root is still present after Step 2, repeat Steps 1 and 2.) Check each proposed solution in the original equation.

4.5 Solving an Equation Involving Square Roots Example Solve Analytic Solution Isolate the radical. Square both sides. Write in standard form and solve.

4.5 Solving an Equation Involving Square Roots These solutions must be checked in the original equation.

4.5 Solving an Equation Involving Square Roots Graphical Solution The equation in the second step of the analytic solution has the same solution set as the original equation. Graph and solve y1 = y2. The only solution is at x = 2.

4.5 Solving an Equation Involving Cube Roots Example Solve Solution

4.5 Solving an Equation Involving Roots (Squaring Twice) Example Solve Solution Isolate radical. Square both sides. Isolate radical. Square both sides. Write in standard quadratic form and solve. A check shows that –1 and 3 are solutions of the original equation.

4.5 Solving Inequalities Involving Rational Exponents Example Solve the inequality Solution The associated equation solution in the previous example was Let Use the x-intercept method to solve this inequality and determine the interval where the graph lies below the x-axis. The solution is the interval .

4.5 Application: Solving a Cable Installation Problem A company wishes to run a utility cable from point A on the shore to an installation at point B on the island (see figure). The island is 6 miles from shore. It costs $400 per mile to run cable on land and $500 per mile underwater. Assume that the cable starts at point A and runs along the shoreline, then angles and runs underwater to the island. Let x represent the distance from C at which the underwater portion of the cable run begins, and the distance between A and C be 9 miles.

4.5 Application: Solving a Cable Installation Problem What are the possible values of x in this problem? Express the cost of laying the cable as a function of x. Find the total cost if 3 miles of cable are on land. Find the point at which the line should begin to angle in order to minimize the total cost. What is this total cost? Solution The value of x must be real where Let k be the length underwater. Using the Pythagorean theorem,

4.5 Application: Solving a Cable Installation Problem The cost of running cable is price  miles. If C is the total cost (in dollars) of laying cable across land and underwater, then If 3 miles of cable are on land, then 3 = 9 – x, giving x = 6.

4.5 Application: Solving a Cable Installation Problem Using the graphing calculator, find the minimum value of y1 = C(x) on the interval (0,9]. The minimum value of the function occurs when x = 8. So 9 – 8 = 1 mile should be along land, and miles underwater. The cost is