ࡱ>  aw)` vbjbj 9{{2&,npnpnp8plr,Js: |"|||yyyȢʢʢʢʢʢʢ$_hǧuyyyy||cuuuy||ȢuyȢuu$|6|>s `$(npCZȟy0ޟ6n,yyuyyyyy jyyyyyyydc*iD*i Chapter 7: Radical Functions and Rational Exponents 7-1 Roots and Radical Expressions nth root for any real numbers a and b, and any positive integer n, if an = b, then a is an nth root of b Ex: 24 = 16 and (-2)4 = 16, both 2 & -2 are fourth roots of 16; -53 = -125, so -5 is the only cube root of -125 ***see chart, p. 369 Ex. #1: Finding all the Real Roots a. Find the cube roots of 0.008 -1000  EMBED Equation.3  b. Find the fourth roots of 1 -0.0001  EMBED Equation.3  More Practice: a. Find all real fifth roots of 0, -1, 32 b. Find all real square roots of 0.0001, -1,  EMBED Equation.3  7-1, p.2 Parts index radical sign   EMBED Equation.COEE2  radicand Principle Root when there are 2 real roots +, the principal root is the positive root; for the (negative) root, then write a - EMBED Equation.3  both roots, write  EMBED Equation.3  Ex. #2: Finding Real Roots a.  EMBED Equation.3  b.  EMBED Equation.3  More Practice: Find each real-number root. a.  EMBED Equation.3  b.  EMBED Equation.3  c.  EMBED Equation.3  nth root of am for any negative real # a,  EMBED Equation.3  when n is even Ex. #3: Simplifying Radical Expressions a.  EMBED Equation.3  b.  EMBED Equation.3  c.  EMBED Equation.3  More Practice: Simplify each radical a.  EMBED Equation.3  b.  EMBED Equation.3  c.  EMBED Equation.3  7-2 Multiplying and Dividing Radical Expressions Multiply Radicals If  EMBED Equation.3  and  EMBED Equation.3  are Real #s, then  EMBED Equation.3 "  EMBED Equation.3  =  EMBED Equation.3  Ex. #1: Multiplying Radicals a.  EMBED Equation.3  b.  EMBED Equation.3  c.  EMBED Equation.3  More Practice: Multiply and simplify. a.  EMBED Equation.3  b.  EMBED Equation.3  c.  EMBED Equation.3  Ex. #2: Simplifying Radical Expressions a.  EMBED Equation.3  b.  EMBED Equation.3  More Practice: Simplify each radical. a.  EMBED Equation.3  b.  EMBED Equation.3  Ex. #3: Multiplying Radical Expressions  EMBED Equation.3  More Practice: Multiply and simplify.  EMBED Equation.3  7-2, p.2 Dividing Radicals If  EMBED Equation.3  and  EMBED Equation.3  are Real #s and b `" 0, then  EMBED Equation.3  Ex. #4: Dividing Radicals a.  EMBED Equation.3  EMBED Equation.3  b.  EMBED Equation.3  More practice: Divide and simplify. a.  EMBED Equation.3  b.  EMBED Equation.3  c.  EMBED Equation.3  Rationalize Denominator rewrite so there are NO radicals in any denominator Ex. #5 Rationalizing the Denominator a.  EMBED Equation.3  b.  EMBED Equation.3  More Practice: Rationalize the denominator a.  EMBED Equation.3  b.  EMBED Equation.3  7-3 Binomial Radical Expressions Like radicals radicals having the same radicand & index Ex. #1: Adding & Subtracting Radical Expressions a.  EMBED Equation.3  b.  EMBED Equation.3  More Practice: Add or subtract. a.  EMBED Equation.3  b.  EMBED Equation.3  c.  EMBED Equation.3  Ex. #3: Simplifying before Adding or Subtracting (a)  EMBED Equation.3  (b)  EMBED Equation.3  More Practice: Simplify.  EMBED Equation.3  7-3 (Part B) Multiplying Radical Expressions Ex. #4: Distributing  EMBED Equation.3  Ex. #5: Multiplying Binomial Radical Expressions  EMBED Equation.3  More Practice: Multiply.  EMBED Equation.3  Conjugates 2 binomials whose first terms are equal and whose last terms are opposites Ex. #6: Multiplying Conjugates  EMBED Equation.3  More Practice: Multiply.  EMBED Equation.3  Ex. #7: Rationalize Binomial Radical Denominators (a)  EMBED Equation.3  (b)  EMBED Equation.3  Extra Practice:  EMBED Equation.3  EMBED Equation.3  Algebra II ~ 7-3 Binomial Radical Expressions Recap Define each term below: Like radicals~ Conjugates~ Add or Subtract if possible.  EMBED Equation.3  2.)  EMBED Equation.3  Simplify. 3.)  EMBED Equation.3  4.)  EMBED Equation.3  Multiply. 5.)  EMBED Equation.3  6.)  EMBED Equation.3  Simplify. Rationalize denominators if necessary. 7.)  EMBED Equation.3  8.)  EMBED Equation.3  9.)  EMBED Equation.3  10.)  EMBED Equation.3  Lesson 7-4 Rational Exponents Rational Exponents If the nth root of a is a real # and m is an integer, then  EMBED Equation.3  Ex. #1: Simplify Expressions w/Rational Exponents a.  EMBED Equation.3  b.  EMBED Equation.3  c.  EMBED Equation.3  More Practice: Simplify a.  EMBED Equation.3  b.  EMBED Equation.3  c.  EMBED Equation.3  Ex. #2: Converting To and From Radical Form a. Write the following in radical form: 1.  EMBED Equation.3  2.  EMBED Equation.3  EMBED Equation.3  b. Write in exponential form: 1.  EMBED Equation.3  2.  EMBED Equation.3  More Practice: Write in radical form: 1.  EMBED Equation.3  2.  EMBED Equation.3  Write in exponential form: 3.  EMBED Equation.3  4.  EMBED Equation.3   Ex. #3: Simplify Numbers w/Rational Exponents a.  EMBED Equation.3  b. 4-3.5 More Practice: Simplify. a.  EMBED Equation.3  b.  EMBED Equation.3  c.  EMBED Equation.3  Ex. #4: Writing Expressions in Simplest form.  EMBED Equation.3  More Practice: Simplify:  EMBED Equation.3  7-5 Solving Radical Equations Review: Solve these simple equations 2x = 10 2x 2 = 10 2 (think, the opposite of multiplying by 2 is dividing by 2) x = _____ x + 3 = 11 x + 3 3 = 11 3 (think, the opposite of adding 3 is subtracting 3) x = _____ x2 = 9  EMBED Equation.3 (Think: the opposite of squaring is taking the square root) Radical Equation an equation having a variable in the radicand or a variable with a rational exponent Ex. #1: Solving Radical Equations with an Index of 2  EMBED Equation.3  More Practice: Solve  EMBED Equation.3  Extraneous Solution a solution that creates a false statement in the original equation (a solution that does not check in the original equation and therefore, it is not considered a solution to the original equation.) Ex. #2: Checking for Extraneous Solutions Solve:  EMBED Equation.3  More Practice: Solve EMBED Equation.3  Solve the equations below. YOU MUST substitute and check each answer. ****Remember to isolate the radical (root), then do the opposite operation. For example, the opposite of  EMBED Equation.3  (cube root) is cubing (cube each side)***** 1.  EMBED Equation.3  CHECK: 2 -  EMBED Equation.3  CHECK: 3.  EMBED Equation.3  CHECK: Ex #3: Solving Radical Equations with Rational Exponents  EMBED Equation.3  More Practice: Solve  EMBED Equation.3  Ex. #4: Solving Equations with Two Rational Exponents Solve: (2x + 1)0.5 (3x + 4)0.5 = 0 More Practice: Solve  EMBED Equation.3  Section 2-1: Relations & Functions Relation: A set of pairs of input and output values. Ex: Suppose a motion detector is used to track an egg as it drops from 10 ft above the ground. The motion detector stores input values (times) and output values (heights). You can write this relation as a set of ordered pairs: Inputs (time in seconds): {0 0.1 0.2 0.3 0.4} Outputs (height in feet): {10 9.8 9.4 8.6 7.4} Relation: { } 1. Graph the relation {(3, 3), (2, 2), (2, 2), (0, 4), (1, 2)}.  Domain: The set of all inputs or x-coordinates of the ordered pairs Range: The set of all outputs or y-coordinates of the ordered pairs 2. Write the ordered pairs for the relation. Find the domain and range.  SHAPE \* MERGEFORMAT  Mapping Diagram: Write the elements of the domain in one region and the elements of the range in another Draw arrows to show how each element from the domain is paired with elements from the range 3. Make a mapping diagram for the following relations. a. {(-1, 7), (1, 3), (1, 7), (-1, 3)} b. {(2, 8), (-1, 5), (0, 8), (-1, 3), (-2, 3)} Function A relation in which each element of the domain is paired with exactly one element of the range. If a relation is a function, then there will be all different x-values in the ordered pairs y-values dont matterthey can be the same or different Discrete functiona collection of isolated points Continuous functionan uninterrupted curve or line 4. Determine whether or not the relation is a function. a. b. {(2,6), (3,6), (4,5), (7,8)} c. {(-1,7), (0,-3), (0,7), ( 1,10)}  SHAPE \* MERGEFORMAT  Vertical Line Test (VLT) You can use the VLT on a graph to tell if a relation is a function If a vertical line drawn on the graph passes through the relation in two or more points, then one element of the domain is paired with more than one element of the range, and it fails the VLT If a relation fails the VLT, it is not a function 5. Use the Vertical Line Test to determine whether each graph is a function. a. b. c.    A function rule expresses an output value in terms of an input value. Examples of function rules: We read function notation  EMBED Equation.3  as  EMBED Equation.3 of  EMBED Equation.3  or a function  EMBED Equation.3 of  EMBED Equation.3 . It does not mean  EMBED Equation.3 times  EMBED Equation.3 !! When the value of  EMBED Equation.3  is 5, we write it as  EMBED Equation.3 , and we read it as  EMBED Equation.3 of 3. 6. Find  EMBED Equation.3  for each function. (a)  EMBED Equation.3  (b)  EMBED Equation.3  (c)  EMBED Equation.3  Practice 2-1 1. Write the ordered pairs for the relation. Find the domain and range.  2. Determine whether the relation {(2, 3), (5, 0), (3, 0), (1, 1)} is a function. 3. Delete one ordered pair so that the relation {(4, 2), (1, 6), (0, 0), (4, 6)} is a function. 4. Determine whether the graph represents a function.  SHAPE \* MERGEFORMAT  5. Find (5) for each function. a.(x) = 5x + 35 b.(x) = x2 x 7-6 Function Operations Function A relation where each element of the domain is paired with exactly one element of range  Ex. #1: Adding and Subtracting Functions Let f(x) = 3x + 8 and g(x) = 2x 12. a. Find f + g b. Find f g More Practice: Let f(x) = 5x2 4x and g(x) = 5x + 1 a. Find f + g b. Find f g Ex. #2: Multiplying and Dividing Functions Let f(x) = x2 1 and g(x) = x + 1 a. Find f " g b. Find  EMBED Equation.3  More Practice: Let f(x) = 6x2 + 7x  5 and g(x) = 2x  1 a. Find f " g b. Find  EMBED Equation.3  Composite Function The output of the first function is the input for the second function: g(f(x)) or (g%f)(x) Ex. #3: Composition of Functions Let f(x) = x  2 and g(x) = x2. a. Find (g%f)(5) More Practice: Evaluate (f%g)(-6) using equations above. Now find (f%g)(x) using the equations above. Ex #4: Let g(x) = x2 + 1 and h(x) = 4x. Find: (g%h)(3) b. (h%g)(-2) c. (g%h)(x) d. 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Find the inverse: b. Graph the relation and its inverse  EMBED Equation.3  More Practice: a. Describe how the line y = x is related to b above b. In a, is the first relation a function? Is its inverse? Ex. #2: Interchanging x and y Find the inverse of y = x2 + 3 More Practice: a. Is y = x2 + 3 a function? Is its inverse? b. Find the inverse of y = 3x 10 Is the inverse a function? Ex. #3: Graphing a Relation and Its Inverse Graph y = x2 + 3 and its inverse  EMBED Equation.3   Inverse of a Function reverse the x with the y in an equation and solve for y (symbol is f -1); if f and f -1 are both functions then they are inverse functions: f -1(f(x)) = f(f -1(x)) = x Ex. #4: Finding an Inverse Function  EMBED Equation.3  a. Find the domain and range. b. Find f -1 c. Find the domain and range for f -1 d. Is a f-1 function? More Practice: Let f(x) = 10 3x. 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