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10.1 Radical Expressions and Functions A radical expression is any expression that can be written in the form π βπ where n is the index (an integer β₯ 2), and a is the radicand. If the index is 2, it is usually not written. The β sign. symbol is called the radical In working with radical expressions, it is necessary to divide them into two groups: those with an even index and those with an odd index. Expressions with an Even Index ο· The radicand must be a nonnegative number (positive or zero). It canβt be negative. The even root of a negative number is βundefinedβ or a βnonreal number.β ββ3 [read βthe square root of negative 3β] is a nonreal number 4 ββ2 [read βthe fourth root of negative 2β] is a nonreal number ο· There are two roots: π The principal (positive) root is represented using β π The negative root is represented using β β If you want both roots, write π β . . π and β β OR π ±β β49 = 7 because 72 = 49 4 β81 = 3 because 34 = 81 ββ81 = β9 because (β9)2 = 81 4 β β16 = β2 because (β2)4 = 16 Expressions with an Odd Index ο· The radicand can be any real number (positive, negative or zero). ο· There is only one root. The root is positive if the radicand is positive. The root is negative if the radicand is negative. π Roots are represented using β . 3 β64 = 4 because 43 = 64 3 ββ64 = β4 because (β4)3 = β64 Radical Functions A radical relation equates a radical expression in one variable to a second (dependent) variable. 3 π¦ = βπ₯ π¦ = β2π₯ + 5 If a radical relation is a radical function, it can be written using function notation. π(π₯) = βπ₯ 3 π(π₯) = β2π₯ + 5 A radical function is written in the form π(π₯) = π΄(π₯ ) where π΄(π₯) is a radical expression with the variable x in the radicand. 3 π(π₯ ) = β5π₯ not a radical function because x is not part of the radicand 3 π(π₯ ) = π₯ β5 is a better way to write this 3 π(π₯ ) = 5 βπ₯ is a radical function π(π‘) = βπ‘ + 3 is a radical function Be careful when identifying the radicand. It is only that part of the expression inside the radical sign. π(π₯ ) = β2x the radicand is 2 Better is π(π₯ ) = π₯β2 π(π₯ ) = β2π₯ the radicand is 2x β(π₯ ) = β2π₯ + 5 the radicand is 2x+5 π(π₯ ) = β2π₯ + 5 the radicand is 2x Domain of a Radical Function The domain of a function is the set of all real-number values for the independent variable that give a realβnumber result. Any value that doesnβt give such a result is excluded from the domain and is referred to as a restricted value. In radical expressions with an odd index, the radicand can be any real number. So, radical functions defined by odd roots have no restricted values. The domain for these functions is all real numbers. {π₯ |π₯ is a real number} or (ββ, β) In radical expressions with an even index, the radicand must be positive or zero. These expressions are not defined for any negative number. So, those real numbers that result in a negative radicand are restricted values and must be excluded from the domain. Finding the Domain Algebraically ο· Set up an inequality of the form radicand β₯ 0 ο· Solve the inequality. ο· Solutions to the inequality are the domain. 4 Example: Find the domain for π(π₯) = β3π₯ + 1. 3π₯ β₯ 0 π₯ β₯ 0 The radicand is 3x. The domain is all real numbers greater than or equal to 0. {π₯| π₯ β₯ 0} or [0, β) -------------------------------------------------------------------------------------Example: Find the domain for β(π₯) = β5π₯ β 4 + 1. The radicand is 5x β 4. 5π₯ β 4 β₯ 0 5π₯ β₯ 4 π₯ β₯ 4/5 4 The domain is all real numbers greater than or equal to . 5 4 {π₯| π₯ β₯ } 5 or 4 [ , β) 5 Example: Find the domain of the function π(π₯) = β2π₯ β 5 + 3 The radicand is 2x β 5. 2π₯ β 5 β₯ 0 2π₯ β₯ 5 π₯β₯ 5 2 The domain is all real numbers greater than or equal to 5 {π₯| π₯ β₯ } or 2 5 2 5 [ , β) 2 -------------------------------------------------------------------------------------Evaluating Radical Functions To evaluate radical functions, substitute the given value for the variable and simplify. Do not leave the expression in radical form. Take the indicated root rounding to two decimal places if necessary. Example: For π (π₯ ) = β5π₯ β 6 , find π(2). π(2) = β5 β 2 β 6 = β10 β 6 = β4 = 2 π(2) = 2 -------------------------------------------------------------------------------------Example: For π(π₯ ) = ββ64 β 8π₯ , find π(β3). π(β3) = ββ64 β 8 β β3 = ββ64 β (β24) = ββ64 + 24 = ββ88 β β9.38 π(β3) β β9.38 -------------------------------------------------------------------------------------3 Example: For π (π₯ ) = βπ₯ β 2 , find π (127). 3 3 π(127) = β127 β 2 = β125 = 5 π (127) = 5 -------------------------------------------------------------------------------------3 Example: For π(π₯ ) = β8π₯ β 8 , find π(β7). 3 3 3 π(β7) = β8 β β7 β 8 = ββ56 β 8 = ββ64 = β4 π(β7) = β4 π Simplifying Radical Expressions of the Form βππ Expressions with n even π βππ = |π| This is the principal ππ‘β root and must be positive. π β βππ = β|π| This is the negative ππ‘β root and must be negative. β(β6)2 = |β6| = 6 β25π₯ 6 = |5π₯ 3 | or 5|x 3 | β(π₯ + 5)2 = |π₯ + 5| 4 β(π₯ β 2)4 = |π₯ β 2| 8 ββ(β5)2 = β|β5| = β5 β βπ¦ 8 = β|π¦| Expressions with n odd π βππ = π The sign of the root depends upon the sign of the radicand. 3 β53 = 5 3 β(β7)3 = β7 5 β(β3)5 = β3 3 β8π₯ 3 = 2π₯