Reviewing Properties of ExponentsExponentiation is repeated Multiplication:To get to each succeeding line, divide by a:a4 = a × a × a × a a3 = a × a × a a2 = a × a a1 = aa0 = 1a-1 = 1/aa-2 = 1/a2

a-3 = 1/a3

Rules of Exponents• We derive the following rules:1) a-n = 1/an

2) am an = am+n

3)

4) (ab)n = an bn

5) (

6) (am)n = amn

Try These

• Simplify the following expressions using only positive exponents:

1) (-3x2)(-4x-2)

2)

3)

Roots and Radical Expressions• If an = b with a and b real numbers and n; a

positive integer, then a is the nth root of b. If n is odd, there is one real nth root of b which we

write as: (We call n, the index, and b; the radicand.)

If n is even, there are two real nth roots of b. The positive root is the principal root, written as: nd the negative root is written as: -.

The only nth root of 0 is 0. For any real number a: = a if n is odd,

= |a| if n is even.

Some Exercises using Radicals

TEXT: Page 365:27, 28, 35, 36, 37

Some Properties of Radicals

1) where and are real.2) where and are real and b≠0.3) Simplest Form: When the radicand with index n has no perfect nth powers as its factors.4) Conjugate Expressions: and , when a and b are

rational, the product of two radical conjugates is rational. Math Teachers often ask students to “rationalize the denominator” so that there are no radicals in the denominator. (Physics teachers don’t ask for this.)

Express the Following in Simplest Form

How Can we Simplify this expression keeping with the Radical notation?

• Hint: Multiply top and bottom by something that makes every factor in the denominator a perfect cube.•

HWK 26

• Text: Page 360: 16, 18, Page 365: 38, 40, 42 Page 371: 14, 26, 36, 40, 48

Binomial Radical Expressions• “Like Radicals” are expressions having the same

index and radicand: are three expressions that are “like radicals”.

• We can add them together using the distributive property:

= = We can apply this property to simplifying more general expressions using radicals.

Simplify the Following Expressions

1)

Simplify These

Another Notation: Rational Exponents

• Find the exponent n that satisfies each of the following equations: so: × so: So In general: and ALL RULES developed for

exponents (slide 2) hold for these fractional exponents.

By extension:

Examples

• Convert to Radical Form:

• Convert to Exponential Form:

HWK 27• Express all answers in simplified form.• Page 378-380: 16, 36, 37, 51, 64, 66• Pages 386-390: 62, 92, 96, 97

Solving Radical Equations

Method is similar to what we did with absolute value equations. Follow these Steps:

1) Isolate Radical; Rational exponent notation may be useful

2) Raise both sides to a power so that unknown will no longer be a radicand.

3) Raising radicals to powers can introduce extraneous solutions. Always check final answers in the original equation.

Solve the Following

1) Solve (4) using the graphing calculator by seeing where LHS and RHS intersect.

Solving Equations with Two Radicals1) Isolate the more complicated radical2) Raise both sides to a power to eliminate the

radical.3) Isolate and eliminate any remaining radical.4) Solve for unknown.5) Check for extraneous solutions.Examples:

HWK 28• Hwk 28 A2T text: Due Thursday 2/11Page 372: 54, 55Pages 395-397: 16, 22, 28, 40, 64

• Hwk 28 A2T text: Due Thursday 2/11Page 372: 54, 55Pages 395-397: 16, 22, 28, 40, 64

• Hwk 28 A2T text: Due Thursday 2/11Page 372: 54, 55Pages 395-397: 16, 22, 28, 40, 64

Function Operations• We can add, subtract, multiply, and divide

functions. We also use a special notation to express these operations:

Some ExamplesPerform the following operations and give the domains of the new functions:1) Let and , find and and state their domains.2) Let and , find and and state their domains.3) Let and , find and and their domains. 4) Let and , find and and their domains.

𝑓

Composite FunctionsWe define the composition of function with function as:

The domain of consists of the x-values in the domain of which produces values of that are in the domain of .Note: The operation is not commutative.

Some Examples

1) Let and . a) Find

b) Find

More Examples2) You have a coupon for $5 off a pizza and a

student ID which gives you 10% off any pizza.

Which results in a cheaper price; applying the coupon then the discount, or vice-versa?

Another Example

3) A store offers a 15% discount on all items and a 20% discount to store employees.

a) Write a model for the price found by taking off the 15% discount before the 20% discount.

b) Write a model for the price found by taking off the 20% discount before the 15% discount.

c) Which results in a cheaper price?

Inverting Relations and FunctionsIf (a,b) is an ordered pair of a relation, then (b,a) is an ordered pair of the relation’s inverse.If both a relation and its inverse are functions, then they are “inverse functions”.The range of a relation is the domain of its inverse.The domain of a relation is the range of its inverse.

The inverse of a function is not necessarily a function:domain range1 1.21 1.42 1.62 1.9

domain range

1.2 1

1.4 1

1.6 2

1.9 2

A function -> Inverse is NOT a function ->

How Can We Invert a Relation? In Tabular Form: Switch the x and y values to get

the inverse in tabular form. In Algebraic Form: Switch the x and y variables in

the original relation to get its inverse. Then solve for y.

Example: Find the inverse of Graph, simultaneously: the function its inverse the line: y=xHow can you graphically find a relation’s inverse?

Notation for the Inverse of a Function

• The inverse of a function f denoted by f -1.(Note: f -1 may not be a function).• Where have we seen this notation before to

denote the inverse of something?

Example

For a) What are the domain and range of ?

b) What is ?

c) What are the domain and range of ?

d) Is a function?

Example

• For a) What are the domain and range of ?

b) What is ?

c) What are the domain and range of ?

d) Is a function?

Inverse of a Formula

• The function: expresses a real relationship.• The inverse of the function is: but it makes no

sense.• So we find the “Inverse of a Formula” by solving

the original relation for the independent variable: .

• Example: Invert to find :

One to One Functions• A function (f) is “one to one” if each y-value in its

range corresponds to exactly one x-value in its domain.

• The inverse of a one to one function is also a function.

• Graphically: A one to one function passes BOTH the vertical line test and the horizontal line test.

• Which functions are one to one?

Composing Inverse Functions• If and are inverse functions, then and for all in the domains of and.Example: For a) Find

b) Find

c) Find

Graphing and Transforming Radical Functions

General Transformations.From a parent function: f(x) we can form the following transformation:g(x) = a f(x-h) + kwhere, the parent function has been: shifted to the right by h THEN scaled by |a| and reflected across the x-axis if

a<0 THEN shifted up by k.

Does Order Matter?• Consider f(x) = x2 and consider the following two series of

transformations:1) Shift to the right by 2, stretch by 3, then shift by 4.

a) Shifting right gives: g1(x) = f(x-2) = (x-2)2

b) Stretch by 3 gives: g2(x) = 3g1(x) = 3(x-2)2

c) Shift up by 4 gives: g3(x) = g2(x)+4 = 3(x-2)2+4

2) Shift to right by 2, shift up by 4, then stretch by 3.a) Shifting right gives: g1(x) = f(x-2) = (x-2)2

b) Shift up by 4 gives: g2(x) = g1(x)+4 = (x-2)2+4

c) Stretch by 3 gives: g3(x) = 3g2(x)= 3(x-2)2+ 12

So: Yes; Pay attention to the order. Each successive transformation acts on the entire function generated by the previous transformation.

Examples• Graph the following using transformation on

the parent functions and check using the graphing calculators:

a) Rewrite in g(x) = a f(x-h) + k form and graph:

Solve by Graphing

• The population of Corpus Christi can be modeled between the years 1970 and 2005 by the radical function: where x is the year.

a) In what year was the population 250000?b) In what year was it 275000?(Hint: Set Window Settings to reasonable values for x and y, then use the intersect function.)

HWK 31

• Due Tuesday 2/24:• Page 419: 53 (to be done in class and added to

hwk 31 for grading.)• Pages 418-419: 36, 44, 48, 54