nth Roots and Rational ExponentsWhat you should learn:Evaluate nth roots of real numbers using both radical notation and rational exponent notationEvaluate the expression.6.1 nth Roots and Rational ExponentsSolving Equations.

6.1 nth Roots and Rational ExponentsEx)Ex) If n is odd, then a has one real nth root: If n is even and a = 0, then a has one nth root:If n is even and a > 0, then a has two real nth roots: If n is even and a < 0, then a has NO Real roots:simplify

6.1 nth Roots and Rational ExponentsEx)Ex)

6.1 nth Roots and Rational ExponentsEx)Ex)

6.1 nth Roots and Rational ExponentsEx)44Ex)55

6.1 nth Roots and Rational ExponentsEx)44Very Important2 answers !Take the Root 1st.

Evaluate the expressions.6.1 nth Roots and Rational Exponents

Properties of Rational Exponents6.2 Properties of Rational Exponents

Review of Properties of Exponents from section 5.1am * an = am+n(am)n = amn(ab)m = ambma-m = = am-n =Page 420 in your book

Ex: Simplify. (no decimal answers)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!c.d.

Ex: Simplify.

== = 5

= = = 2Ex: Write the expression in simplest form.

= = =

=

= =

=Cant 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..9672.828

Ex: Perform the indicated operation5(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.

6.2 Rational Exponents and Radical FunctionsSimplify the ExpressionsEx)Ex)Ex)Ex)

More Examplesa.

b.

c.

d.

Directions: Simplify the Expression. Assume all variables are positive.a.

(16g4h2)1/2 = 161/2g4/2h2/2 = 4g2h

c. d.

6.2 Properties of Rational ExponentsYes, Change both to rational exponent form and use the quotient property.

6.3 Power Functions and Functions OperationsA2.2.5

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)Division h(x) = f(x)/g(x)Composition h(x) = f(g(x)) OR g(f(x))** Domain all real x-values that make sense (i.e. cant have a zero in the denominator, cant take the even nth root of a negative number, etc.)

Example: Let f(x) = 3x1/3 & g(x) = 2x1/3Find (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 numbersThe SUMThe DIFFERENCE

Ex: Let f(x) = 4x1/3 & g(x) = x1/2. Find (a) the product, (b) the quotient, and (c) the domain for each.a.)The PRODUCTb.)The QUOTIENT

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 = Domain of (a) all reals 0, because you cant take the 6th root of a negative number.Domain of (b) all reals > 0, because you cant take the 6th root of a negative number and you cant have a denominator of zero.

Compositionf(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.Perform Function Operations and CompositionCont

You purchase a baseball glove with a price tag of $180 dollars. The sports store applies a newspaper coupon of $50 and a 10% store discount. When the coupon is applied before the discount. When the discount is applied before the coupon.ABWe will visit this question later

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 =(b) (2x-1)2-1 = 22x-2-1 = (c) 2(2x-1)-1 = 2(2-1x) =(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 cant have x = 0.g(f(x))f(f(x))f(g(x))

You purchase a baseball glove with a price tag of $180 dollars. The sports store applies a newspaper coupon of $50 and a 10% store discount. Find the final price of the purchase when the coupon is applied before the discount.Find the final price when the discount is applied before the coupon.AB

STEP 1 Write functions for the discounts.Function for $50 coupon: f(x) = x - 50 Function for 10% discount: g(x) = x 0.10x = 0.90xSTEP 2 Compose the functions$50 coupon is applied first: g(f(x)) = g(x 50) = 0.90(x - 50)10% discount is applied first: f(g(x)) = f(0.90x) = 0.90x - 50

STEP 3 Evaluate the functions g(f(x)) f(g(x)) when x = 180 $50 coupon is applied first:g(f(180)) = 0.90(180 50) = $11710% discount is applied first: f(g(180)) = 0.90(180) 50 = $112The final price is $117 when the $50 coupon is applied before the 10% discount.The final price is $112 when the 10% discount is applied before the $50 coupon.

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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.6.3 Power Functions and Functions Operations

Inverse FunctionsWhat you should learn:Find inverses of linear functions.Verify that f and g are inverse functions.6.4 Inverse FunctionsMichigan Standard A2.2.6Graph the function f. Then use the graph to determine whether the inverse of f is a function.

Temperature ConversionThe formula to convert temperatures from degrees Celsius-to-Fahrenheit isBut, how do you convert from Fahrenheit-to-Celsius?? C =Toon in later

Review from chapter 2Relation a mapping of input values (x-values) onto output values (y-values).Here are 3 ways to show the same relation.y = x2 x y-2 4 -1 10 01 1EquationTable of valuesGraph

Inverse relation just think: switch the x & y-values.x = y2x y -2 -10 01 1* 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 f(x) = -3x+6. -switch x & y -solve for yy = -3x + 6x = -3y + 6 x - 6 = -3y- change f(x) with y3 Steps:

Verify Inverse FunctionsGiven 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)= -4x+8 and g(x) = -1/4x + 2 are inverses.Meaning find f(g(x)) and g(f(x)). If they both equal x, then they are inverses.f(g(x))= -4(-1/4x + 2) + 8= x 8 + 8= xg(f(x))= -1/4(-4x + 8) + 2= x 2 + 2= x ** Because f(g(x))= x and g(f(x)) = x, they are inverses.

Temperature ConversionThe formula to convert temperatures from degrees Celsius-to-Fahrenheit isBut, how do you convert from Fahrenheit-to-Celsius?? C =

Temperature ConversionSolve for C- 32- 32

Ex: (a) Find the inverse of f(x) = x5.y = x5x = y5 (b) Is f -1(x) a function?(hint: look at the graph!Does it pass the vertical line test?)Yes , f -1(x) is a function.

Horizontal Line TestUsed to determine whether a functions 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.f -1(x) is not a function.y = 2x2 - 4x = 2y2 - 4x + 4 = 2y2OR, if you fix the tent in the basement

Ex: g(x)=2x3Inverse is a function!y = 2x3x = 2y3OR, if you fix the tent in the basement

State the domain and range of the function.1)2)All real #s

Solving Radical EquationsWhat you should learn:Solve equations that contain Radicals.6.6 Solving Radical EquationsSolve equations that contain Rational exponents.Michigan Standard L1.2.1

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

6.6 Solving Radical EquationsEx.2)Key Step: Before raising each side to the same power, you should isolate the radical expression on one side of the equation.Simple Radical

6.6 Solving Radical EquationsOne RadicalEx.3)

6.6 Solving Radical EquationsTwo RadicalsEx.4)

6.6 Solving Radical EquationsRadicals with an Extraneous SolutionWhat 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 EquationsRadicals with an Extraneous SolutionEx.5)

6.6 Solving Radical EquationsRadicals with an Extraneous SolutionEx.5)

Without solving, explain why 6.6 Solving Radical Equationshas no solution.

6.6 Solving Radical EquationsSolve equations that contain Rational exponents.Ex. 6)it

6.6 Solving Radical EquationsSolve equations that contain Rational exponents.Ex. 7)it

Without solving, explain why 6.6 Solving Radical Equationshas no solution.