Inverse Functions Inverse Relations The inverse of a
![Inverse Functions Inverse Functions](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-1.jpg)
![Inverse Relations The inverse of a relation is the set of ordered pairs obtained Inverse Relations The inverse of a relation is the set of ordered pairs obtained](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-2.jpg)
![Tables and Graphs of Inverses Switch x and y Orginal (0, 25) (2, 16) Tables and Graphs of Inverses Switch x and y Orginal (0, 25) (2, 16)](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-3.jpg)
![Inverse and Compositions In order for two functions to be inverses: AND Inverse and Compositions In order for two functions to be inverses: AND](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-4.jpg)
![One-to-One Functions A function f(x) is one-to-one on a domain D if, for every One-to-One Functions A function f(x) is one-to-one on a domain D if, for every](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-5.jpg)
![The Horizontal Line Test If a horizontal line intersects a curve more than once, The Horizontal Line Test If a horizontal line intersects a curve more than once,](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-6.jpg)
![The Horizontal Line Test If a horizontal line intersects a curve more than once, The Horizontal Line Test If a horizontal line intersects a curve more than once,](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-7.jpg)
![Example Without graphing, decide if the function below has an inverse function. If f Example Without graphing, decide if the function below has an inverse function. If f](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-8.jpg)
![Find the Inverse of a Function 1. Switch the x and y of the Find the Inverse of a Function 1. Switch the x and y of the](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-9.jpg)
![Example Find the inverse of the following: Only Half Parabola Switch x and y Example Find the inverse of the following: Only Half Parabola Switch x and y](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-10.jpg)
![Logarithms v Exponentials Logarithms v Exponentials](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-11.jpg)
![Definition of Logarithm The logarithm base a of b is the exponent you put Definition of Logarithm The logarithm base a of b is the exponent you put](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-12.jpg)
![Logarithm and Exponential Forms Logarithm Form 5 = log 2(32) Logs Give you Exponents Logarithm and Exponential Forms Logarithm Form 5 = log 2(32) Logs Give you Exponents](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-13.jpg)
![Examples Write each equation in exponential form 1. log 125(25) = 2/3 1252/3 = Examples Write each equation in exponential form 1. log 125(25) = 2/3 1252/3 =](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-14.jpg)
![Example Complete the table if a is a positive real number and: Domain All Example Complete the table if a is a positive real number and: Domain All](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-15.jpg)
![The Change of Base Formula The following formula allows you to evaluate any valid The Change of Base Formula The following formula allows you to evaluate any valid](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-16.jpg)
![Solving Equations with the Change of Base Formula Solve: Isolate the base and power Solving Equations with the Change of Base Formula Solve: Isolate the base and power](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-17.jpg)
![Properties of Logarithms For a>0, b>0, m≠ 1, and any real number n. Logarithm Properties of Logarithms For a>0, b>0, m≠ 1, and any real number n. Logarithm](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-18.jpg)
![Example 1 Condense the expression: Example 1 Condense the expression:](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-19.jpg)
![Example 2 Expand the expression: Example 2 Expand the expression:](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-20.jpg)
![Example 3 Solve the equation: Example 3 Solve the equation:](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-21.jpg)
![AP Reminders Do not forget the following relationships: AP Reminders Do not forget the following relationships:](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-22.jpg)
![Inverse Trigonometry Inverse Trigonometry](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-23.jpg)
![Tangent Cosine Each one of these trigonometric functions fail the horizontal line test, so Tangent Cosine Each one of these trigonometric functions fail the horizontal line test, so](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-24.jpg)
![Tangent Cosine In order for their inverses to be functions, the domains of the Tangent Cosine In order for their inverses to be functions, the domains of the](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-25.jpg)
![Trigonometric Functions with Restricted Domains Function f (x) = sin x f (x) = Trigonometric Functions with Restricted Domains Function f (x) = sin x f (x) =](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-26.jpg)
![Cos-1 Sec-1 Cot-1 Tan-1 Csc-1 Sin-1 Inverse Trigonometric Functions Cos-1 Sec-1 Cot-1 Tan-1 Csc-1 Sin-1 Inverse Trigonometric Functions](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-27.jpg)
![Inverse Trigonometric Functions Function f (x) = sin-1 x f (x) = cos-1 x Inverse Trigonometric Functions Function f (x) = sin-1 x f (x) = cos-1 x](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-28.jpg)
![Alternate Names/Defintions for Inverse Trigonometric Functions Familiar f (x) = sin-1 x -1 f Alternate Names/Defintions for Inverse Trigonometric Functions Familiar f (x) = sin-1 x -1 f](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-29.jpg)
![Example 1 Evaluate: This expression asks us to find the angle whose sine is Example 1 Evaluate: This expression asks us to find the angle whose sine is](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-30.jpg)
![Example 2 Evaluate: This expression asks us to find the angle whose cosecant is Example 2 Evaluate: This expression asks us to find the angle whose cosecant is](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-31.jpg)
![Example 3 Evaluate: The embedded expression asks us to find the angle whose sine Example 3 Evaluate: The embedded expression asks us to find the angle whose sine](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-32.jpg)
![Example 4 Evaluate: The embedded expression asks us to find the angle whose cosine Example 4 Evaluate: The embedded expression asks us to find the angle whose cosine](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-33.jpg)
![Example 3 Evaluate: The embedded expression asks us to find the angle whose tangent Example 3 Evaluate: The embedded expression asks us to find the angle whose tangent](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-34.jpg)
![White Board Challenge Evaluate without a calculator: White Board Challenge Evaluate without a calculator:](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-35.jpg)
![White Board Challenge Evaluate without a calculator: White Board Challenge Evaluate without a calculator:](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-36.jpg)
![White Board Challenge Evaluate without a calculator: White Board Challenge Evaluate without a calculator:](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-37.jpg)
![White Board Challenge Evaluate without a calculator: White Board Challenge Evaluate without a calculator:](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-38.jpg)
![White Board Challenge Evaluate without a calculator: White Board Challenge Evaluate without a calculator:](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-39.jpg)
- Slides: 39
![Inverse Functions Inverse Functions](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-1.jpg)
Inverse Functions
![Inverse Relations The inverse of a relation is the set of ordered pairs obtained Inverse Relations The inverse of a relation is the set of ordered pairs obtained](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-2.jpg)
Inverse Relations The inverse of a relation is the set of ordered pairs obtained by switching the input with the output of each ordered pair in the original relation. (The domain of the original is the range of the inverse; and vice versa) Ex: and are inverses because their input and output are switched. For instance: 22 4
![Tables and Graphs of Inverses Switch x and y Orginal 0 25 2 16 Tables and Graphs of Inverses Switch x and y Orginal (0, 25) (2, 16)](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-3.jpg)
Tables and Graphs of Inverses Switch x and y Orginal (0, 25) (2, 16) (18, 16) (6, 4) (14, 4) (10, 0) X 0 2 6 10 14 18 20 Y 25 16 4 0 4 16 25 X 25 16 4 0 4 16 25 Y 0 2 6 10 14 18 20 Inverse Switch x and y (16, 18) (4, 14) (0, 10) (4, 6) (16, 2) Although transformed, the graphs are identical Line of Symmetry: y = x
![Inverse and Compositions In order for two functions to be inverses AND Inverse and Compositions In order for two functions to be inverses: AND](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-4.jpg)
Inverse and Compositions In order for two functions to be inverses: AND
![OnetoOne Functions A function fx is onetoone on a domain D if for every One-to-One Functions A function f(x) is one-to-one on a domain D if, for every](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-5.jpg)
One-to-One Functions A function f(x) is one-to-one on a domain D if, for every value c, the equation f(x) = c has at most one solution for every x in D. Or, for every a and b in D: Theorems: 1. A function has an inverse function if and only if it is one-to-one. 2. If f is strictly monotonic (strictly increasing or decreasing) on its entire domain, then it is one-to-one and therefore has an inverse function.
![The Horizontal Line Test If a horizontal line intersects a curve more than once The Horizontal Line Test If a horizontal line intersects a curve more than once,](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-6.jpg)
The Horizontal Line Test If a horizontal line intersects a curve more than once, it’s inverse is not a function. Use the horizontal line test to decide which graphs have an inverse that is a function. Make sure to circle the functions.
![The Horizontal Line Test If a horizontal line intersects a curve more than once The Horizontal Line Test If a horizontal line intersects a curve more than once,](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-7.jpg)
The Horizontal Line Test If a horizontal line intersects a curve more than once, it’s inverse is not a function. Use the horizontal line test to decide which graphs have an inverse that is a function. Make sure to circle the functions.
![Example Without graphing decide if the function below has an inverse function If f Example Without graphing, decide if the function below has an inverse function. If f](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-8.jpg)
Example Without graphing, decide if the function below has an inverse function. If f is strictly monotonic (strictly increasing or decreasing) on its entire domain, then it is one-to-one and therefore has an inverse function. See if the derivative is always one sign: Since the derivative is always negative, the inverse of f is a function.
![Find the Inverse of a Function 1 Switch the x and y of the Find the Inverse of a Function 1. Switch the x and y of the](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-9.jpg)
Find the Inverse of a Function 1. Switch the x and y of the function whose inverse you desire. 2. Solve for y to get the Inverse function 3. Make sure that the domains and ranges of your inverse and original function match up.
![Example Find the inverse of the following Only Half Parabola Switch x and y Example Find the inverse of the following: Only Half Parabola Switch x and y](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-10.jpg)
Example Find the inverse of the following: Only Half Parabola Switch x and y Really y = Solve for y Full Parabola (too much) Restrict the Domain! x=3 Make sure to check with a table and graph on the calculator.
![Logarithms v Exponentials Logarithms v Exponentials](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-11.jpg)
Logarithms v Exponentials
![Definition of Logarithm The logarithm base a of b is the exponent you put Definition of Logarithm The logarithm base a of b is the exponent you put](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-12.jpg)
Definition of Logarithm The logarithm base a of b is the exponent you put on a to get b: a>0 and b>0 i. e. Logs give you exponents! The logarithm to the base e, denoted ln x, is called the natural logarithm.
![Logarithm and Exponential Forms Logarithm Form 5 log 232 Logs Give you Exponents Logarithm and Exponential Forms Logarithm Form 5 = log 2(32) Logs Give you Exponents](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-13.jpg)
Logarithm and Exponential Forms Logarithm Form 5 = log 2(32) Logs Give you Exponents Input Becomes Output Base Stays the Base 5 2 = 32 Exponential Form
![Examples Write each equation in exponential form 1 log 12525 23 12523 Examples Write each equation in exponential form 1. log 125(25) = 2/3 1252/3 =](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-14.jpg)
Examples Write each equation in exponential form 1. log 125(25) = 2/3 1252/3 = 25 2. Log 8(x) = 1/3 81/3 = x Write each equation in logarithmic form 1. If 64 = 4 3 2. If 1/27 = 3 x log 4(64) = 3 Log 3(1/27) = x
![Example Complete the table if a is a positive real number and Domain All Example Complete the table if a is a positive real number and: Domain All](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-15.jpg)
Example Complete the table if a is a positive real number and: Domain All Reals All Positive Reals Range All Positive Reals All Reals Continuous? Yes One-to-One? Yes Always Up Always Down Concavity Left End Behavior Right End Behavior
![The Change of Base Formula The following formula allows you to evaluate any valid The Change of Base Formula The following formula allows you to evaluate any valid](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-16.jpg)
The Change of Base Formula The following formula allows you to evaluate any valid logarithm statement: For a and b greater than 0 AND b≠ 1. Example: Evaluate
![Solving Equations with the Change of Base Formula Solve Isolate the base and power Solving Equations with the Change of Base Formula Solve: Isolate the base and power](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-17.jpg)
Solving Equations with the Change of Base Formula Solve: Isolate the base and power Change the exponential equation to an logarithm equation Use the Change of Base Formula
![Properties of Logarithms For a0 b0 m 1 and any real number n Logarithm Properties of Logarithms For a>0, b>0, m≠ 1, and any real number n. Logarithm](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-18.jpg)
Properties of Logarithms For a>0, b>0, m≠ 1, and any real number n. Logarithm of 1: Logarithm of the base: Power Property: Product Property: Quotient Property:
![Example 1 Condense the expression Example 1 Condense the expression:](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-19.jpg)
Example 1 Condense the expression:
![Example 2 Expand the expression Example 2 Expand the expression:](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-20.jpg)
Example 2 Expand the expression:
![Example 3 Solve the equation Example 3 Solve the equation:](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-21.jpg)
Example 3 Solve the equation:
![AP Reminders Do not forget the following relationships AP Reminders Do not forget the following relationships:](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-22.jpg)
AP Reminders Do not forget the following relationships:
![Inverse Trigonometry Inverse Trigonometry](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-23.jpg)
Inverse Trigonometry
![Tangent Cosine Each one of these trigonometric functions fail the horizontal line test so Tangent Cosine Each one of these trigonometric functions fail the horizontal line test, so](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-24.jpg)
Tangent Cosine Each one of these trigonometric functions fail the horizontal line test, so they are not one-to-one. Therefore, there inverses are not functions. Cosecant Secant Cotangent Sine Trigonometric Functions
![Tangent Cosine In order for their inverses to be functions the domains of the Tangent Cosine In order for their inverses to be functions, the domains of the](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-25.jpg)
Tangent Cosine In order for their inverses to be functions, the domains of the trigonometric functions are restricted so that they become oneto-one. Cosecant Secant Cotangent Sine Trigonometric Functions with Restricted Domains
![Trigonometric Functions with Restricted Domains Function f x sin x f x Trigonometric Functions with Restricted Domains Function f (x) = sin x f (x) =](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-26.jpg)
Trigonometric Functions with Restricted Domains Function f (x) = sin x f (x) = cos x f (x) = tan x f (x) = csc x f (x) = sec x f (x) = cot x Domain Range
![Cos1 Sec1 Cot1 Tan1 Csc1 Sin1 Inverse Trigonometric Functions Cos-1 Sec-1 Cot-1 Tan-1 Csc-1 Sin-1 Inverse Trigonometric Functions](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-27.jpg)
Cos-1 Sec-1 Cot-1 Tan-1 Csc-1 Sin-1 Inverse Trigonometric Functions
![Inverse Trigonometric Functions Function f x sin1 x f x cos1 x Inverse Trigonometric Functions Function f (x) = sin-1 x f (x) = cos-1 x](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-28.jpg)
Inverse Trigonometric Functions Function f (x) = sin-1 x f (x) = cos-1 x -1 f (x) = tan x f (x) = csc-1 x -1 f (x) = sec x -1 f (x) = cot x Domain Range
![Alternate NamesDefintions for Inverse Trigonometric Functions Familiar f x sin1 x 1 f Alternate Names/Defintions for Inverse Trigonometric Functions Familiar f (x) = sin-1 x -1 f](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-29.jpg)
Alternate Names/Defintions for Inverse Trigonometric Functions Familiar f (x) = sin-1 x -1 f (x) = cos x -1 f (x) = tan x f (x) = csc-1 x -1 f (x) = sec x -1 f (x) = cot x Alternate Calculator f (x) = arcsin x f (x) = sin-1 x -1 f (x) = arccos x f (x) = cos x -1 f (x) = arctan x f (x) = arccsc x f (x) = sin-1 1/x -1 f (x) = arcsec x f (x) = cos 1/x -1 f (x) = arccot x f (x) = -tan x+ Arccot is different because it is always positive but tan can be negative.
![Example 1 Evaluate This expression asks us to find the angle whose sine is Example 1 Evaluate: This expression asks us to find the angle whose sine is](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-30.jpg)
Example 1 Evaluate: This expression asks us to find the angle whose sine is ½. Remember the range of the inverse of sine is.
![Example 2 Evaluate This expression asks us to find the angle whose cosecant is Example 2 Evaluate: This expression asks us to find the angle whose cosecant is](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-31.jpg)
Example 2 Evaluate: This expression asks us to find the angle whose cosecant is -1 (or sine is -1). Remember the range of the inverse of cosecant is.
![Example 3 Evaluate The embedded expression asks us to find the angle whose sine Example 3 Evaluate: The embedded expression asks us to find the angle whose sine](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-32.jpg)
Example 3 Evaluate: The embedded expression asks us to find the angle whose sine is 1/3. Draw a picture (There are infinite varieties): It does not even matter what the angle is, we only need to find: Find the missing side length(s) Is the result positive or negative?
![Example 4 Evaluate The embedded expression asks us to find the angle whose cosine Example 4 Evaluate: The embedded expression asks us to find the angle whose cosine](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-33.jpg)
Example 4 Evaluate: The embedded expression asks us to find the angle whose cosine is -1/6. Ignore the Draw a picture (There are infinite varieties): negative for now. It does not even matter what the angle is, we only need to find: Find the missing side length(s) Is the result positive or negative?
![Example 3 Evaluate The embedded expression asks us to find the angle whose tangent Example 3 Evaluate: The embedded expression asks us to find the angle whose tangent](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-34.jpg)
Example 3 Evaluate: The embedded expression asks us to find the angle whose tangent is x. Draw a possible picture (There are infinite varieties): It does not even matter what the angle is, we only need to find: Find the missing side length(s) Is the result positive or negative?
![White Board Challenge Evaluate without a calculator White Board Challenge Evaluate without a calculator:](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-35.jpg)
White Board Challenge Evaluate without a calculator:
![White Board Challenge Evaluate without a calculator White Board Challenge Evaluate without a calculator:](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-36.jpg)
White Board Challenge Evaluate without a calculator:
![White Board Challenge Evaluate without a calculator White Board Challenge Evaluate without a calculator:](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-37.jpg)
White Board Challenge Evaluate without a calculator:
![White Board Challenge Evaluate without a calculator White Board Challenge Evaluate without a calculator:](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-38.jpg)
White Board Challenge Evaluate without a calculator:
![White Board Challenge Evaluate without a calculator White Board Challenge Evaluate without a calculator:](https://slidetodoc.com/presentation_image_h/04dc33ba4e1ee39a7e05742e97025343/image-39.jpg)
White Board Challenge Evaluate without a calculator:
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