Unit 7: Radical Functions & Rational Exponents

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1 Date Period Unit 7: Radical Functions & Rational Exponents DAY 0 TOPIC Roots and Radical Expressions Multiplying and Dividing Radical Expressions Binomial Radical Expressions Rational Exponents 4 Solving Radical Equations 5 Function Operations 6 Inverse Relations and Functions 7 Graphs of Radical Functions 8 Review

2 U7D0: Roots and Radical Expressions (7.) Read pages 6 65 in your textbook to prepare yourself to do the homework problems. All of this information should be review from a previous course. The only new topic is the idea of absolute value. Do the best you can with this we will go over it to start class tomorrow. Here are some additional notes that might help you U7D: Multiplying & Dividing Radical Expressions (7.) Vocab: Like radicals have Think of this the same was as you think about like terms Conjugates Page of 8

3 NOTES THROUGH EXAMPLES: Page of 8

4 U7D: Binomial Radical Expressions (7.) Page of 8

5 Page 4 of 8

6 U7D: Rational Exponents (7.4) Property Equals Example a m a a n m n ab m m a a a m n a b x a m x y a Page 5 of 8

7 U7D4: Solving Radical Equations (7.4) A radical equation is an equation that has a variable in a radicand or has a variable with a rational exponent. ( x ) x 0 5 radical equations x 0 NOT a radical equation Give your own: Radical equation Non radical equation To solve a radical equation: isolate the radical on one side of the equation and then raise both sides of the equation to the same power.. x 6. 5x 6 0 CHECK!! You can solve equations of the form m n x n k by raising each side of the equation to the power. m. x x 54 Page 6 of 8

8 Extraneous solutions can be introduced when you raise both sides of an equation to a power. So CHECK all possible solutions in the original equation! 5. x 5 x 6. 5 x x CHECK x x x x 7 0 CHECK Closure: investigate checking for extraneous solutions using your calculator.. Solve x x 7 5. How many apparent solutions do you get?. Any or all apparent solutions mabe extraneous. One way to find out is to let y = the left side of the equation and y = the right sides. Graph the two equations. In how many points do they intersect?. The x-values of the points of intersection are solutions of the original equation. Are any or the apparent solutions extraneous? 4. Substitute the apparent solutions in the original equation. Does this algebraic check agree with the calculator check? Page 7 of 8

9 U7D5: Function Operations (7.6) Function Operations Operation Function Operation Examples ( f g)( x) f( x) g( x) f x 5x 4 x g x 5x Addition Subtraction ( f g)( x) f( x) g( x) Domain: ( f g)() f x 5x 4 x g x 5x Domain: ( f g)() ( f g) x f( x) g( x) f x x g x x Multiplication Division f f( x) ( x), g( x) 0 g g( x) Domain: ( f g) 4 f x x g x x f Domain: (6) g Page 8 of 8

10 Definition of a composition function: f g x f g x f of g of x g f x g f x g of f of x Example: f x 5x g x x Page 9 of 8

11 More practice with composite functions: Now, evaluating composite functions: WOrDie: Additional Practice: Page 0 of 8

12 U7D6: Inverse Relations & Functions (7.7) Warmup: What do the words relation and function mean? How do they compare? How do you determine if a relation is a function? Use the graphs below An inverse function the process that a function does Relation r DOMAIN RANGE Relation s DOMAIN RANGE Plot each ordered pair given below. Then, write the inverse point by switching x, y to yx., Point, 4,, 5,, x, y Inverse yx, Extension: What line is each point reflected over? Page of 8

13 To find the inverse of a function, rewrite with x and y, then switch the letters and solve for y Example : y x Example : f x x 0 Domain/Range of Function: Domain/Range of Function: Domain/Range of Inverse: Domain/Range of Inverse: Is the inverse a function? Is the inverse a function? Find f (this is the notation for the of f ) when f x 4x. Then find f f 6. FACT: Composition of Inverse functions If f and f are inverse functions, then f f x and f f x. Examples: For the function f x 4 7 x, find f f 9 and f f. Given the relation y x, answer the following. a) Is the relation a function? b) What is the domain of the relation? c) What is the range of the relation? d) What is the inverse of the relation? e) What is the domain of the inverse? f) What is the range of the inverse? g) Graph the relation and use the line of symmetry to graph the inverse is it a function?! Page of 8

14 Note: The inverse of a function MAY NOT be a function. If the inverse is also a function, it is referred to as the inverse function. Method - Determine graphically if a function has an inverse which is also a function: Use the horizontal line test to determine if a function has an inverse function. If ANY horizontal line intersects your original function in ONLY ONE location, your function has an inverse which is also a function. The function y = x +, shown at the right, HAS an inverse function because it passes the horizontal line test. Method - Determine graphically if a function has an inverse which is also a function: If a function has an inverse function, the reflection of that original function in the identity line y = x will also be a function (it will pass the vertical line test for functions). The example at the left shows the original function, y = x, in blue solid. The reflection over the identity line y = x is shown in red - dashed. The red dashed line will not pass the vertical line test for functions, thus y = x does not have an inverse function. You can see that the inverse exists, but it is NOT a function. NOTE: With functions such as y = x, it is possible to restrict the domain to obtain an inverse function for a portion of the graph. This means that you will be looking at only a selected section of the original graph that will pass the horizontal line test for the existence of an inverse function. For example: or } by restricting the graph in such a manner, you guarantee the existence of an inverse function for a portion of the graph. (Other restrictions are also possible.) Page of 8

15 Additional Practice Wrap up: What is an inverse of a function? How do you find an inverse from a graph/equation? Page 4 of 8

16 U7D7: Graphs of Radical Functions (7.8) Two Types of Radical Functions Function Type Square Root Cube Root Parent Function y x y x Equation with Parameters y a xh k y a xh k My Personal Notes Page 5 of 8

17 U7D8: Review!. If f(x)= x and g(x)=x, what is g f ( x)? a. 6x b. 9x c. 8x d. 4 8x. Which expression is NOT equivalent to a. xy b x y 4 x y c. 4 x y d. 4 9x y. Which expression is NOT equivalent to 4 4n a. 4 (4 n ) b. n c. n d. n 4. How do you write with a rationalized denominator? 5 a. - b. c. 4 d If f( x) 4x, a. 4 what is f ( f (4)? b. 0 c. 4 d How can you write a. 5 xy b. 0 xy 5 xy with a rationalized denominator c. 0x y xy d. 4 x y xy 7. Which of the following expressions is in simplified form? a. 0x b. 8x c. 6 d. y= 5 8. Which statement is NOT true? 4 a. 9 b. 9 c. 7 d. 8 Page 6 of 8

18 9. If f(x)= x and g(x)=x, what is g f ( x)? 4 a. 8x b. 8x c. 6x d. 9x 0 Which expression is NOT equivalent to a. 4 4n b. 4 4n n c. n d. n. Simplify the expression completely. CHECK for extraneous solutions ( points each) x b. 4 a. 4 x 7. Simplify the expression completely. CHECK for extraneous solutions a. x 8 b. x 7 x Page 7 of 8

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