nth Roots and Rational Exponents 6.1 nth Roots and Rational Exponents What you should learn: Goal 1 Evaluate nth roots of real numbers using both radical notation and rational exponent notation Goal 2 Evaluate the expression. Goal 3 Solving Equations. 6.1 nth Roots and Rational Exponents

Using Rational Exponent Notation Goal Using Rational Exponent Notation Rewrite the expression using RATIONAL EXPONENT notation. 1 If n is odd, then a has one real nth root: Ex) If n is even and a > 0, then a has two real nth roots: Ex) If n is even and a = 0, then a has one nth root: If n is even and a < 0, then a has NO Real roots: 6.1 nth Roots and Rational Exponents

Using Rational Exponent Notation Rewrite the expression using RADICAL notation. Ex) Ex) 6.1 nth Roots and Rational Exponents

Evaluating Expressions Goal Evaluating Expressions Evaluate the expression. 2 Ex) Ex) 6.1 nth Roots and Rational Exponents

Solving Equations Goal 3 Ex) Ex) 4 4 Ex) Ex) 5 5 When the exponent is EVEN you must use the Plus/Minus When the exponent is ODD you don’t use the Plus/Minus 6.1 nth Roots and Rational Exponents

Solving Equations Take the Square 1st. Very Important 2 answers ! Ex) 4 4 Take the Square 1st. Ex) Very Important 2 answers ! 6.1 nth Roots and Rational Exponents

Assignment Pg. 404 # 13 – 61 odd Reflection on the Section Evaluate the expressions. Assignment Pg. 404 # 13 – 61 odd 6.1 nth Roots and Rational Exponents

Properties of Rational Exponents 6.2 Properties of Rational Exponents What you should learn: Goal 1 Use properties of rational exponents to evaluate and simplify expressions. Use properties of rational exponents to solve real-life problems. Goal 2 6.2 Properties of Rational Exponents

Review of Properties of Exponents from section 6.1 am * an = am+n (am)n = amn (ab)m = ambm a-m = = am-n = These all work for fraction exponents as well as integer exponents.

Ex: Simplify. (no decimal answers) (43 * 23)-1/3 = (43)-1/3 * (23)-1/3 = 4-1 * 2-1 = ¼ * ½ = 1/8 d. = = = 61/2 * 61/3 = 61/2 + 1/3 = 63/6 + 2/6 = 65/6 b. (271/3 * 61/4)2 = (271/3)2 * (61/4)2 = (3)2 * 62/4 = 9 * 61/2 ** All of these examples were in rational exponent form to begin with, so the answers should be in the same form!

Ex: Write the expression in simplest form. Ex: Simplify. = = = 5 = = 2 Ex: Write the expression in simplest form. = = = = = Can’t have a Radical in the basement! ** If the problem is in radical form to begin with, the answer should be in radical form as well.

Ex: Perform the indicated operation 5(43/4) – 3(43/4) = 2(43/4) b. = c. = If the original problem is in radical form, the answer should be in radical form as well. If the problem is in rational exponent form, the answer should be in rational exponent form.

More Examples a. b. c. d.

Ex: Simplify the Expression. Assume all variables are positive. (16g4h2)1/2 = 161/2g4/2h2/2 = 4g2h c. d.

Assignment Pg. Reflection on the Section Can you find the quotient of two radicals with different indices? Yes, Change both to rational exponent form and use the quotient property. Assignment Pg. 6.2 Properties of Rational Exponents

Perform Function Operations and Composition 6.3 Perform Function Operations and Composition What you should learn: Goal 1 Perform operations with functions including power functions. Goal 2 Use power functions and function operations to solve real-life problems. A2.2.5 6.3 Power Functions and Functions Operations

Operations on Functions: for any two functions f(x) & g(x) Addition h(x) = f(x) + g(x) Subtraction h(x) = f(x) – g(x) Multiplication h(x) = f(x)*g(x) OR f(x)g(x) Division h(x) = f(x)/g(x) OR f(x) ÷ g(x) Composition h(x) = f(g(x)) OR g(f(x)) ** Domain – all real x-values that “make sense” (i.e. can’t have a zero in the denominator, can’t take the even nth root of a negative number, etc.)

Domain of (a) all real numbers Domain of (b) all real numbers Ex: Let f(x)=3x1/3 & g(x)=2x1/3. Find (a) the sum, (b) the difference, and (c) the domain for each. 3x1/3 + 2x1/3 = 5x1/3 3x1/3 – 2x1/3 = x1/3 Domain of (a) all real numbers Domain of (b) all real numbers

Ex: Let f(x)=4x1/3 & g(x)=x1/2 Ex: Let f(x)=4x1/3 & g(x)=x1/2. Find (a) the product, (b) the quotient, and (c) the domain for each. 4x1/3 * x1/2 = 4x1/3+1/2 = 4x5/6 = 4x1/3-1/2 = 4x-1/6 = (c) Domain of (a) all reals ≥ 0, because you can’t take the 6th root of a negative number. Domain of (b) all reals > 0, because you can’t take the 6th root of a negative number and you can’t have a denominator of zero.

Composition f(g(x)) means you take the function g and plug it in for the x-values in the function f, then simplify. g(f(x)) means you take the function f and plug it in for the x-values in the function g, then simplify.

Ex: Let f(x)=2x-1 & g(x)=x2-1 Ex: Let f(x)=2x-1 & g(x)=x2-1. Find (a) f(g(x)), (b) g(f(x)), (c) f(f(x)), and (d) the domain of each. (a) 2(x2-1)-1 = (c) 2(2x-1)-1 = 2(2-1x) = (b) (2x-1)2-1 = 22x-2-1 = (d) Domain of (a) all reals except x=±1. Domain of (b) all reals except x=0. Domain of (c) all reals except x=0, because 2x-1 can’t have x=0.

Reflection on the Section How is the composition of functions different form the product of functions? The composition of functions is a function of a function. The output of one function becomes the input of the other function. The product of functions is the product of the output of each function when you multiply the two functions. Assignment Page # 6.3 Power Functions and Functions Operations

Inverse Functions 6.4 What you should learn: Goal 1 Find inverses of linear functions. Goal 2 Verify that f and g are inverse functions. Goal 3 Graph the function f. Then use the graph to determine whether the inverse of f is a function. Michigan Standard A2.2.6 6.4 Inverse Functions

Review from chapter 2 x y -2 4 y = x2 -1 1 0 0 1 1 Relation – a mapping of input values (x-values) onto output values (y-values). Here are 3 ways to show the same relation. x y -2 4 -1 1 0 0 1 1 y = x2 Equation Table of values Graph

Inverse relation – just think: switch the x & y-values. -2 -1 0 0 1 1 x = y2 ** the inverse of an equation: switch the x & y and solve for y. ** the inverse of a table: switch the x & y. ** the inverse of a graph: the reflection of the original graph in the line y = x.

To find the inverse of a function: Change the f(x) to a y. Switch the x & y values. Solve the new equation for y. ** Remember functions have to pass the vertical line test!

Ex: Find an inverse of y = -3x+6. Steps: -switch x & y -solve for y y = -3x + 6 x = -3y + 6 x - 6 = -3y

Inverse Functions Given 2 functions, f(x) & g(x), if f(g(x)) = x AND g(f(x)) = x, then f(x) & g(x) are inverses of each other. Symbols: f -1(x) means “f inverse of x”

Ex: Verify that f(x)= -3x+6 and g(x) = -1/3x + 2 are inverses. Meaning find f(g(x)) and g(f(x)). If they both equal x, then they are inverses. f(g(x))= -3(-1/3x + 2) + 6 = x – 6 + 6 = x g(f(x))= -1/3(-3x + 6) + 2 = x – 2 + 2 = x ** Because f(g(x))= x and g(f(x)) = x, they are inverses.

Ex: (a) Find the inverse of f(x) = x5. (b) Is f -1(x) a function? (hint: look at the graph! Does it pass the vertical line test?) y = x5 x = y5 Yes , f -1(x) is a function.

Horizontal Line Test Used to determine whether a function’s inverse will be a function by seeing if the original function passes the horizontal line test. If the original function passes the horizontal line test, then its inverse is a function. If the original function does not pass the horizontal line test, then its inverse is not a function.

Ex: Graph the function f(x)=x2 and determine whether its inverse is a function. Graph does not pass the horizontal line test, therefore the inverse is not a function.

Ex: f(x)=2x2 - 4 Determine whether f -1(x) is a function, then find the inverse equation. y = 2x2 - 4 x = 2y2 - 4 x + 4 = 2y2 OR, if you fix the tent in the basement… f -1(x) is not a function.

Ex: g(x)=2x3 y = 2x3 x = 2y3 Inverse is a function! OR, if you fix the tent in the basement… Inverse is a function!

assignment Reflection on the Section Describe the steps for finding the inverse of a relation. Write the original equation; switch x and y; solve for y assignment 6.4 Inverse Functions

Graphing Square Root and Cube Root Functions. 6.5 Graphing Square Root and Cube Root Functions. What you should learn: Goal 1 Graph square root and cube root functions. Goal 2 Use square root and cube root functions to solve real-life problems. Michigan Standard A2.3.3 6.5 Graphing Square Root and Cube Root Functions

Domain and range: all real numbers Graphing Radical Functions You have seen the graphs of y = x and y = x . These are examples of radical functions. 3 Domain and range: all real numbers Domain: x ³ 0, Range: y ³ 0

Domain and range: all real numbers Graphing Radical Functions You have seen the graphs of y = x and y = x . These are examples of radical functions. 3 Domain: x ³ 0, Range: y ³ 0 Domain and range: all real numbers In this lesson you will learn to graph functions of the form y = a x – h + k and y = a x – h + k. 3

Graphing Radical Functions GRAPHS OF RADICAL FUNCTIONS To graph y = a x – h + k or y = a x – h + k, follow these steps. 3 Sketch the graph of y = a x or y = a x . 3 1 STEP 2 STEP Shift the graph h units horizontally and k units vertically.

Comparing Two Graphs Describe how to obtain the graph of y = x + 1 – 3 from the graph of y = x . SOLUTION Note that y = x + 1 – 3 = x – (–1) + (–3), so h = –1 and k = –3. To obtain the graph of y = x + 1 – 3, shift the graph of y = x left 1 unit and down 3 units.

Graphing a Square Root Function Graph y = –3 x – 2 + 1. SOLUTION 1 Sketch the graph of y = –3 x (shown dashed). Notice that it begins at the origin and passes through the point (1, –3). 2 Note that for y = –3 x – 2 + 1, h = 2 and k = 1. So, shift the graph right 2 units and up 1 unit. The result is a graph that starts at (2, 1) and passes through the point (3, –2).

Graphing a Cube Root Function Graph y = 3 x + 2 – 1. 3 SOLUTION Sketch the graph of y = 3 x (shown dashed). Notice that it passes through the origin and the points (–1, –3) and (1, 3). 3 1 Note that for y = 3 x + 2 – 1, h = –2 and k = –1. 3 2 So, shift the graph left 2 units and down 1 unit. The result is a graph that passes through the points (–3, –4), (–2, –1), and (–1, 2).

Finding Domain and Range State the domain and range of the functions in the previous examples. SOLUTION From the graph of y = –3 x – 2 + 1, you can see that the domain of the function is x ³ 2 and the range of the function is y £ 1.

Finding Domain and Range State the domain and range of the functions in the previous examples. SOLUTION From the graph of y = –3 x – 2 + 1, you can see that the domain of the function is x ³ 2 and the range of the function is y £ 1. SOLUTION From the graph of y = 3 x + 2 – 1, you can see that the domain and range of the function are both all real numbers. 3

Using Radical Functions in Real Life AMUSEMENT PARKS At an amusement park a ride called the rotor is a cylindrical room that spins around. The riders stand against the circular wall. When the rotor reaches the necessary speed, the floor drops out and the centrifugal force keeps the riders pinned to the wall. When you use radical functions in real life, the domain is understood to be restricted to the values that make sense in the real-life situation. The model that gives the speed s (in meters per second) necessary to keep a person pinned to the wall is s = 4.95 r where r is the radius (in meters) of the rotor. Use a graphing calculator to graph the model. Then use the graph to estimate the radius of a rotor that spins at a speed of 8 meters per second.

Modeling with a Square Root Function AMUSEMENT PARKS At an amusement park a ride called the rotor is a cylindrical room that spins around. The riders stand against the circular wall. When the rotor reaches the necessary speed, the floor drops out and the centrifugal force keeps the riders pinned to the wall. SOLUTION Graph y = 4.95 x and y = 8. Choose a viewing window that shows the point where the graphs intersect. Then use the Intersect feature to find the x-coordinate of that point. You get x  2.61. The radius is about 2.61 meters.

Modeling with a Cube Root Function Biologists have discovered that the shoulder height h (in centimeters) of a male African elephant can be modeled by h = 62.5 t + 75.8 3 where t is the age (in years) of the elephant. Use a graphing calculator to graph the model. Then use the graph to estimate the age of an elephant whose shoulder height is 200 centimeters. SOLUTION Graph y = 62.5 x + 75.8 and y = 200 with your calculator. Choose a viewing window that shows the point where the graphs intersect. Then use the Intersect feature to find the x-coordinate of that point. 3 You get x  7.85 The elephant is about 8 years old.

assignment Reflection on the Section Give an example of a radical function. assignment 7.5 Graphing Square Root and Cube Root Functions

Solve equations that contain Radicals. 6.6 Solving Radical Equations What you should learn: Goal 1 Solve equations that contain Radicals. Solve equations that contain Rational exponents. Goal 2 Michigan Standard L1.2.1 6.6 Solving Radical Equations

Solve equations that contain Radicals Goal 1 Solve equations that contain Radicals Solve the equation. Check for extraneous solutions. Key Step: To raise each side of the equation to the same power. Simple Radical Ex.1) check your solutions!! 6.6 Solving Radical Equations

Don’t forget to check it. Simple Radical Ex.2) Key Step: Before raising each side to the same power, you should isolate the radical expression on one side of the equation. Don’t forget to check it. 6.6 Solving Radical Equations

Don’t forget to check your solutions!! One Radical Ex.3) Don’t forget to check your solutions!! 6.6 Solving Radical Equations

Don’t forget to check your solutions!! Two Radicals Ex.4) Don’t forget to check your solutions!! 6.6 Solving Radical Equations

Radicals with an Extraneous Solution What is an Extraneous Solution? … is a solution to an equation raised to a power that is not a solution to the original equation. Example 5) 6.6 Solving Radical Equations

Ex.5) Radicals with an Extraneous Solution 6.6 Solving Radical Equations

Don’t forget to check your solutions!! Radicals with an Extraneous Solution Ex.5) Don’t forget to check your solutions!! 6.6 Solving Radical Equations

Goal 2 it Ex. 6) Solve equations that contain Rational exponents. 6.6 Solving Radical Equations

Let’s try … Solving the Rational Exponent equation using the TI-84. Example) Y= Y= 4

Reflection on the Section Without solving, explain why has no solution. Assignment Page 441 17 - 53 odd 6.6 Solving Radical Equations