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STANDARDS OF LEARNING
CONTENT REVIEW NOTES
ALGEBRA II
2nd Nine Weeks, 2018-2019
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OVERVIEW
Algebra II Content Review Notes are designed by the High School Mathematics Steering Committee as a
resource for students and parents. Each nine weeksโ Standards of Learning (SOLs) have been identified and a
detailed explanation of the specific SOL is provided. Specific notes have also been included in this document
to assist students in understanding the concepts. Sample problems allow the students to see step-by-step models
for solving various types of problems. A โ โ section has also been developed to provide students with the
opportunity to solve similar problems and check their answers. Supplemental online information can be
accessed by scanning QR codes throughout the document. These will take students to video tutorials and online
resources. In addition, a self-assessment is available at the end of the document to allow students to check their
readiness for the nine-weeks test. It is the Mathematics Instructorsโ desire that students and parents will use this
document as a tool toward the studentsโ success on the end-of-year assessment.
The document is a compilation of information found in the Virginia Department of Education (VDOE)
Curriculum Framework, Enhanced Scope and Sequence, and Released Test items. In addition to VDOE
information, Prentice Hall Textbook Series and resources have been used. Finally, information from various
websites is included. The websites are listed with the information as it appears in the document.
Supplemental online information can be accessed by scanning QR codes throughout the document. These will
take students to video tutorials and online resources. In addition, a self-assessment is available at the end of the
document to allow students to check their readiness for the nine-weeks test.
The Algebra II Blueprint Summary Table is listed below as a snapshot of the reporting categories, the number
of questions per reporting category, and the corresponding SOLs.
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The Quadratic Formula and the Discriminant
Sometimes you will need to solve a quadratic that cannot be factored. In that case you
can use the quadratic formula: ๐ฅ =โ๐ยฑโ๐2โ4๐๐
2๐
You just substitute the values for a, b, and c into the quadratic formula and simplify.
Example 1: Solve 5๐ฅ2 โ 2๐ฅ โ 9 = 0
๐ = 5 ๐ = โ2 ๐ = โ9 Plug these values into the quadratic formula
๐ฅ =โ๐ยฑโ๐2โ4๐๐
2๐ ๐ฅ =
2ยฑโ(โ2)2โ4(5)(โ9)
2(5) ๐ฅ =
2ยฑโ4+180
10 ๐ฅ =
2ยฑโ184
10
Your two solutions are ๐ฅ =2+โ184
10 and ๐ฅ =
2โโ184
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In Example 7, the quadratic had two solutions, but a quadratic can also have only one
solution or no solutions at all.
You can determine how many solutions a quadratic equation will have by calculating
the discriminant (๐2 โ 4๐๐). This is the part that is located under the square root in the
quadratic formula. If the discriminant is positive, the quadratic will have two real
solutions. If the discriminant is zero, the quadratic will have one real solution. If the
discriminant is negative, the quadratic will have no real solutions.
Two Solutions (the parabola
crosses the x-axis twice)
One Solution (the parabola
crosses the x-axis one time)
No Solutions (the parabola does
not cross the x-axis)
Scan this QR code to go to a video tutorial on finding the
solutions of a quadratic.
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Example 2: Determine the number of solutions of 7๐ฅ2 โ ๐ฅ = 12
Start by getting one side equal to zero and write in standard form.
7๐ฅ2 โ ๐ฅ = 12
โ12 โ 12
7๐ฅ2 โ ๐ฅ โ 12 = 0 Now, find the discriminant.
๐ = 7 ๐ = โ1 ๐ = โ12
๐2 โ 4๐๐ = (โ1)2 โ 4(7)(โ12) = 1 + 336 = 337
Because the discriminant is positive, this quadratic has two solutions.
Quadratic Functions
1. Find the roots 9๐ฅ2 + 48๐ฅ + 64 = 0
2. Determine the number of solutions 4๐ฅ2 โ 3๐ฅ + 10 = 0
3. Find the zeros ๐ฅ2 โ 5๐ฅ = 14
4. Find the solutions โ2๐ฅ2 + 7 = โ4๐ฅ
Completing the Square
AII.3 The student will solve b) quadratic equations over the set of complex numbers;
Completing the Square is one way to solve quadratic equations. When you complete
the square, you will create a perfect square trinomial which can then be factored. In
order to complete the square, you first need to transform the equation into the form
๐ฅ2 + ๐๐ฅ = ๐. This means that you will get all of the variable terms on one side, and the
constant on the other. Also, be sure that the coefficient of the ๐ฅ2 term is 1.
Next you will โcomplete the squareโ by adding ( ๐
2 )2 to each side of the equation.
Then, you can factor the trinomial. Take the square root of both sides and solve for ๐ฅ.
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Example 1: Solve by completing the square: ๐ฅ2 + 8 = 8๐ฅ
๐ฅ2 + 8 = 8๐ฅ
โ8 โ 8
๐ฅ2 = 8๐ฅ โ 8
โ8๐ฅ โ 8๐ฅ
๐ฅ2 โ 8๐ฅ = โ8
๐ฅ2 โ 8๐ฅ + 16 = โ8 + 16
(๐ฅ โ 4)2 = 8
๐ฅ โ 4 = ยฑโ8
๐ฅ = 4 ยฑ 2 โ2
Example 2: Solve by completing the square: 3๐ฅ2 + 6๐ฅ โ 18 = 0
3๐ฅ2 + 6๐ฅ โ 18 = 0
+18 + 18
3๐ฅ2 + 6๐ฅ = 18
รท 3 รท 3
๐ฅ2 + 2๐ฅ = 6
๐ฅ2 + 2๐ฅ + 1 = 6 + 1
(๐ฅ + 1)2 = 7
๐ฅ + 1 = ยฑโ7
โ1 โ 1
๐ฅ = โ1 ยฑ โ7
Transform the equation into the form
๐ฅ2 + ๐๐ฅ = ๐.
Add ( ๐
2 )2 to both sides of the
equation.
๐ = โ8 so ( โ8
2 )2
( โ8
2 )2 = (โ4)2 = 16
Factor the left side, and simplify the right.
Take the square root of both sides
Solve for x. Donโt forget to simplify your
square root if you can!!
Scan this QR code to go to a video tutorial on completing
the square.
Transform the equation into the form
๐ฅ2 + ๐๐ฅ = ๐.
Divide by 3 to get the coefficient of ๐ฅ2 to 1. Remember that you are dividing all terms by 3!
Add ( ๐
2 )2 to both sides of the
equation.
๐ = 2 so ( 2
2 )2
( 2
2 )2 = (1)2 = 1
Factor the left side, and simplify the right.
Take the square root of both sides
Solve for x. Donโt forget to simplify your
square root if you can!!
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Complex Numbers
AII.2 The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i.
AII.3 The student will solve a) absolute value linear equations and inequalities;
b)quadratic equations over the set of complex numbers;
The complex numbers are made up of the real numbers and the imaginary numbers.
Complex Numbers (๐ + ๐๐)
The imaginary number, i, is the same as โโ1 ๐ = โโ1 ๐กโ๐๐๐๐๐๐๐ ๐2 = โ1
Example 1: Simplify โโ8
We can start by pulling ๐ out to factor the negative out from under the radical.
๐ โ โ8
Now simplify the radical by pulling out any perfect square factors (i.e. 4).
๐ โ โ4 โ 2
2๐ โ2
You can also perform operations on complex numbers. Remember that if you get an ๐2
term that it can be replaced with โ1.
Example 2: Simplify (3 โ 3๐) โ (โ4 + 5๐) Donโt forget to distribute the negative!!
3 โ 3๐ + 4 โ 5๐ And then combine like terms!
7 โ 8๐
Real Numbers
(๐ + 0๐)
Imaginary Numbers
(๐ + ๐๐, ๐ โ 0)
Pure Imaginary
Numbers (0 +๐๐, ๐ โ 0)
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Example 3: Simplify (3 โ 3๐)(โ4 + 5๐) You can FOIL this!
โ12 + 15๐ + 12๐ โ 15๐2 Combine like terms
โ12 + 27๐ โ 15๐2 Replace ๐2 with -1
โ12 + 27๐ โ 15(โ1) Simplify
โ12 + 27๐ + 15
3 + 27๐
Number pairs that are of the form ๐ + ๐๐ and ๐ โ ๐๐ are called complex conjugates.
When you multiply complex conjugates, the product is a real number. This can be used
to simplify quotients of complex numbers.
Example 4: Simplify (9 โ 5๐)(9 + 5๐)
81 + 45๐ โ 45๐ โ 25๐2 Notice that the ๐ terms cancel
81 โ 25๐2 Replace ๐2 with -1
81 โ 25(โ1)
81 + 25
106
Example 5: Find the quotient by using the complex conjugate.
8โ16๐
4๐ Multiply the numerator and denominator by โ4๐
8โ16๐
4๐โ
โ4๐
โ4๐
โ32๐+64๐2
โ16๐2 Replace ๐2 with โ1
โ32๐+64(โ1)
โ16(โ1)
โ64โ32๐
16= โ4 โ 2๐ Simplify!
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Complex Numbers Simplify
1. โโ32๐ฅ
2. (6 โ 4๐) + (2 โ 3๐) โ (6 โ 6๐)
3. โ5๐(4 + 3๐)
4. (2 + ๐)(3 โ 2๐)
5. 6โ10๐
โ2๐
6. 4+7๐
1โ3๐
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Polynomial Functions
AII.4 The student will solve systems of linear-quadratic and quadratic-quadratic equations, algebraically and graphically.
AII.6 For absolute value, square root, cube root, rational, polynomial, exponential, and logarithmic functions, the student will a) recognize the general shape of function families; and b) use knowledge of transformations to convert between equations and the
corresponding graphs of functions. AII.7 The student will investigate and analyze linear, quadratic, absolute value,
square root, cube root, rational, polynomial, exponential, and logarithmic function families algebraically and graphically. Key concepts include b) intervals in which a function is increasing or decreasing; h) end behavior;
AII.8 The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression.
Systems of Equations The solution to a system of equations is the place or places where two equations intersect. These two equations can be linear, quadratic, or a combination of both. Systems of two linear equations can have no solutions, one solution, or infinitely many solutions.
No Solutions One Solution Infinitely Many Solutions
Two lines that are parallel.
These lines have the same slope, but different y-intercepts.
They will never intersect. Therefore, there is no solution.
Two lines that intersect.
These lines have different slopes, which causes them to
intersect in one place. Therefore, there is one solution. In this example, the solution is
(2, 3).
Two lines that are the same.
These lines have the same slope and the same y-intercept. This means they are the same line
and will share all points. Therefore, there are infinitely
many solutions.
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Systems of one linear and one quadratic equation can have no solutions, one solution, or two solutions.
No Solutions One Solution Two Solutions
Because these two graphs never intersect, this system
of equations has no solution.
The line and the parabola share one common point. This system of equations
has one solution.
The line intersects the parabola in two places.
This system of equations has two solutions.
Systems of equations can be solved graphically or algebraically. To solve a system graphically, you will graph both equations and determine the intersection points. This can be verified on your calculator.
Scan this QR code to go to a video tutorial on solving
systems of linear equations.
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Example 1: Solve this system of equations by graphing.
๐ฆ = ๐ฅ2 + 5๐ฅ โ 4
๐ฆ = ๐ฅ โ 4
First graph both equations in your calculator.
Next, push the button, then the button. This will take you to the
CALCULATE menu.
Your calculator will then ask you โSecond curve?โ Again scroll your cursor near the
intersection point on the second graph and press enter. It will then ask you to โGuess?โ
an intersection point. This is unnecessary, just hit enter again, and it will display the
intersection point at the bottom of the screen.
The solution to this system of equations is (-4, -8) and (0, -4).
You can see that the line crosses the parabola in two places which means this system will have two solutions! We will use the Calculate: Intersection function of your calculator to determine the ordered pairs.
2ND TRACE
You will select 5: intersect
Your calculator will ask you โFirst curve?โ Scroll your cursor near the first
intersection point and push Enter.
The two graphs intersect at the point (-4, -8).
The two graphs intersect at another
point as well. Repeat the above procedure to find the other
intersection point.
Scan this QR code to go to a video tutorial on solving systems
of linear and quadratic equations.
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To solve a system of equations algebraically, you will use the substitution method. We will solve the same system that we just solved graphically, algebraically now. Example 1: Solve this system of equations algebraically.
๐ฆ = ๐ฅ2 + 5๐ฅ โ 4
๐ฆ = ๐ฅ โ 4
The substitution property states that you can substitute two equal quantities in an equation. In our system of equations, we can substitute (๐ฅ โ 4) for ๐ฆ from the second equation, into the first equation. This substitution will give us:
๐ฅ โ 4 = ๐ฅ2 + 5๐ฅ โ 4
โ๐ฅ + 4 โ ๐ฅ + 4 0 = ๐ฅ2 + 4๐ฅ 0 = ๐ฅ (๐ฅ + 4) ๐ฆ = 0 โ 4 = โ4 (0, โ4)
๐ฆ = โ4 โ 4 = โ8 (โ4, โ8)
Polynomial Functions
1. How many solutions does this system have? ๐ฆ = 7๐ฅ โ 8 ๐๐๐ ๐ฆ = 3๐ฅ2 โ 2๐ฅ + 4
2. Solve the system using either method. ๐ฆ = 2๐ฅ + 1 ๐๐๐ ๐ฆ = ๐ฅ2 โ 4๐ฅ + 9
3. Solve the system by graphing, round your answers to the nearest hundredth.
๐ฆ = ๐ฅ2 + 2๐ฅ โ 7 ๐๐๐ ๐ฆ = โ2๐ฅ2 + 4๐ฅ + 3
Move terms so that one side of the equation is equal to 0, then factor to solve!
You can now see that ๐ฅ = 0 or ๐ฅ = โ4. Plug these two solutions into one of the original equations to find the corresponding y-values.
This will give you your ordered pairs.
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Polynomials Graphs
Degree 0 1 2
Name Constant Linear Quadratic
Polynomial Example
๐ฆ = โ2 ๐ฆ = ๐ฅ + 3 ๐ฆ = 1
2๐ฅ2 + 3๐ฅ
Graph Example
Degree 3 4 5
Name Cubic Quartic Quintic
Polynomial Example
๐ฆ = 1
4๐ฅ3 โ 2 ๐ฆ = ๐ฅ4 โ 3๐ฅ3 + 2๐ฅ2 + 2
๐ฆ = โ๐ฅ5 + ๐ฅ (*note: graph does not
match equation*)
Graph Example
End Behavior: This describes what the graph does on each end. It is usually described as either going up or down.
As an example, this graph is going down to the left and up to the right.
Relative Minimum and Maximum: This describes any โpeaksโ or โvalleysโ in your
graph. Absolute Minimum and Maximum: This describes the lowest and highest points on
the graph.
This can be expressed as: ๐(๐ฅ) โ โโ ๐๐ ๐ฅ โ โโ
๐(๐ฅ) โ โ๐๐ ๐ฅ โ โ
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You may be asked to state where the graph is increasing or decreasing. You will do this in interval notation, reading the graph from left to right.
This graph is decreasing over the intervals: (โโ, ๐ด) โช (๐ต, ๐ถ)
This graph is increasing over the intervals: (๐ด, ๐ต) โช (๐ถ, โ)
The zeros of a graph refer to places on the graph where ๐ฆ = 0. These will be your
x-intercepts. This graph appears to have 3 zeros at approximately โ2.3, 0, ๐๐๐ 1.3.
Domain and Range refer to all of the x-values (domain) and y-values (range) that a
graph will have if you extend the ends as far as they will go. When determining the
domain, think about the x-axis, and when thinking about range consider the y-axis.
In this graph, there is a relative maximum at B, and relative minimums at A and C. A would also be described as the absolute minimum of this graph.
B is NOT the absolute maximum.
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In the graph pictured below, even though the graph is traveling to the right and left very
slowly, along the x-axis, it will ultimately cross all x-coordinates from โโ to โ. Thus
the domain of this graph would be (โโ, โ).
The range of this graph refers to all of the y-values that this graph will ultimately touch.
This graph is coming down from the left and going up to the right which means that it
has an absolute minimum point, on this graph. That point is A. This graph will never
have a y-value smaller than the y-value at point A (which is approximately โ5.1).
Thus, the range of this graph is [โ๐. ๐, โ). This means that the range does include
โ5.1 because that is where point A is, and continues all the way up to infinity.
Polynomial Functions
4. For the polynomial function shown below state the:
Scan this QR code to go to a video tutorial on describing
polynomial functions.
a) end behavior
b) the intervals where it is
increasing and decreasing
c) the relative and absolute
minimums and maximums
d) zeros
e) domain and range
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Composition and Inverse Functions
AII.7 The student will investigate and analyze linear, quadratic, absolute value, square root, cube root, rational, polynomial, exponential, and logarithmic function families algebraically and graphically. Key concepts include j) inverse of a function; and k) composition of functions algebraically and graphically.
Function Operations
Addition (๐ + ๐)(๐ฅ) = ๐(๐ฅ) + ๐(๐ฅ)
Subtraction (๐ โ ๐)(๐ฅ) = ๐(๐ฅ) โ ๐(๐ฅ)
Multiplication (๐ โ ๐)(๐ฅ) = ๐(๐ฅ) โ ๐(๐ฅ)
Division (๐
๐) (๐ฅ) =
๐(๐ฅ)
๐(๐ฅ) ๐(๐ฅ) โ 0
Example 1: Let ๐(๐ฅ) = ๐ฅ2 โ 3๐ฅ ๐๐๐ ๐(๐ฅ) = 2๐ฅ2 โ 5.
Find (๐ + ๐)(๐ฅ) and (๐ โ ๐)(๐ฅ). State the domain of each.
(๐ + ๐)(๐ฅ) = ๐(๐ฅ) + ๐(๐ฅ) = ๐ฅ2 โ 3๐ฅ + (2๐ฅ2 โ 5)
๐ฅ2 โ 3๐ฅ + 2๐ฅ2 โ 5 = 3๐ฅ2 โ 3๐ฅ โ 5
Because the domain of both original functions is all real numbers, the domain of the
sum is also all real numbers, (โโ, โ).
(๐ โ ๐)(๐ฅ) = ๐(๐ฅ) โ ๐(๐ฅ) = ๐ฅ2 โ 3๐ฅ โ (2๐ฅ2 โ 5)
๐ฅ2 โ 3๐ฅ โ 2๐ฅ2 + 5 = โ๐ฅ2 โ 3๐ฅ + 5
Because the domain of both original functions is all real numbers, the domain of the
difference is also all real numbers, (โโ, โ).
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Example 2: Let ๐(๐ฅ) = ๐ฅ โ 2 ๐๐๐ ๐(๐ฅ) = ๐ฅ2 โ 4.
Find (๐ โ ๐)(๐ฅ) and (๐
๐)(๐ฅ). State the domain of each.
(๐ โ ๐)(๐ฅ) = ๐(๐ฅ) โ ๐(๐ฅ) = (๐ฅ โ 2)(๐ฅ2 โ 4)
๐ฅ3 โ 2๐ฅ2 โ 4๐ฅ + 8
Because the domain of both original functions is all real numbers, the domain of the
product is also all real numbers, (โโ, โ).
Example 2: continued
(๐
๐) (๐ฅ) =
๐(๐ฅ)
๐(๐ฅ)=
๐ฅโ2
๐ฅ2โ4
๐ฅ โ 2
(๐ฅ + 2)(๐ฅ โ 2)
1
๐ฅ + 2
Because any value that would make the denominator 0 is not in the domain of this
function, we need to determine these values before stating the domain.
๐ฅ2 โ 4 = 0 โซ (๐ฅ + 2)(๐ฅ โ 2) = 0 โซ ๐ฅ = 2, โ2
The values of 2 and -2 would make the denominator equal to 0. Therefore, those
values are not in the domain. The domain is (โโ, โ2) โช (โ2, 2) โช (2, โ).
Composition Functions
(๐ โ ๐)(๐ฅ) = ๐(๐(๐ฅ))
First, you will evaluate ๐(๐ฅ). Then, you will substitute that value in for ๐ฅ in ๐(๐ฅ).
Factor the denominator to see if you can simplify!
๐ฅโ2
๐ฅโ2 divide to make 1!
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Example 3: Let ๐(๐ฅ) = ๐ฅ + 7 ๐๐๐ ๐(๐ฅ) = 3๐ฅ2 โ 1.
Find (๐ โ ๐)(1) and (๐ โ ๐)(1).
(๐ โ ๐)(1) = ๐(๐(1))
๐(1) = 3(1)2 โ 1 = (3 โ 1) โ 1 = 3 โ 1 = 2
๐(2) = 2 + 7 = 9
(๐ โ ๐)(1) = ๐(๐(1))
๐(1) = 1 + 7 = 8
๐(8) = 3(8)2 โ 1 = 3 โ 64 โ 1 = 192 โ 1 = 191
Inverse Functions If an original function has (๐ฅ, ๐ฆ) as an ordered pair, this functionโs inverse will have (๐ฆ, ๐ฅ) as an ordered pair.
Thus to find the inverse of a function, you simply switch the ๐ฅ and ๐ฆ. A functionโs
inverse is often written as ๐โ1(๐ฅ).
Example 4: What is the inverse of the relation listed below?
x f(x)
3 11
5 23
7 74
Example 5: Given ๐(๐ฅ) = ๐ฅ2 โ 5 find ๐โ1(๐ฅ). In order to find the inverse, switch the x and y terms, and then transform the function for y.
๐ฆ = ๐ฅ2 โ 5
๐ฅ = ๐ฆ2 โ 5
+5 + 5
๐ฅ + 5 = ๐ฆ2
ยฑโ๐ฅ + 5 = ๐ฆ
x f(x)
11 3
23 5
74 7
Scan this QR code to go to a video tutorial on composition
and inverse functions.
The inverse can be founding by switching the x and y terms.
Inverse
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Composition and Inverse Functions
Given ๐(๐ฅ) = ๐ฅ โ 4 ๐(๐ฅ) = ๐ฅ2 โ(๐ฅ) = ๐ฅ2 โ ๐ฅ โ 12
1. Find (๐ + โ)(๐ฅ)
2. Find (โ โ ๐)(๐ฅ)
3. Find (๐ โ ๐)(๐ฅ)
4. Find ( โ
๐ )(๐ฅ)
5. Find (โ โ ๐)(4)
6. Find (๐ โ โ)(3)
7. Find ๐โ1(๐ฅ) ๐๐๐ ๐โ1(๐ฅ)
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Answers to the problems:
Quadratic Functions
1. ๐ฅ = โ8
3
2. Zero 3. ๐ฅ = โ2, 7 4. ๐ฅ โ โ1.121, 3.121 Complex Numbers
1. 4๐โ2๐ฅ 2. 2 โ ๐ 3. 15 โ 20๐ 4. 8 โ ๐ 5. 5 + 3๐
6. โ17+19๐
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Polynomial Functions 1. ๐๐ ๐ ๐๐๐ข๐ก๐๐๐ 2. (2, 5) ๐๐๐ (4, 9) 3. (โ1.52, โ7.73) ๐๐๐ (2.19, 2.17)
4. ๐. ๐(๐ฅ) โ โโ ๐๐ ๐ฅ โ โโ ๐(๐ฅ) โ โโ ๐๐ ๐ฅ โ โ
๐. ๐ผ๐๐๐๐๐๐ ๐๐๐ (โโ, โ2.7) โช (0.3, 2.6) ๐ท๐๐๐๐๐๐ ๐๐๐ (โ2.7, 0.3) โช (2.6, โ)
๐. ๐ ๐๐๐๐ก๐๐ฃ๐ ๐๐๐ฅ: 6 ๐คโ๐๐ ๐ฅ = โ2.7
0.67 ๐คโ๐๐ ๐ฅ = 2.6
๐ด๐๐ ๐๐๐ข๐ก๐ ๐๐๐ฅ: 6 ๐คโ๐๐ ๐ฅ = โ2.7
๐ ๐๐๐๐ก๐๐ฃ๐ ๐๐๐: โ 2.7 ๐คโ๐๐ ๐ฅ = 0.3
๐ด๐๐ ๐๐๐ข๐ก๐ ๐๐๐: ๐ท๐๐๐ ๐๐๐ก ๐๐ฅ๐๐ ๐ก
๐. โ4, โ1, 2, 3
๐. ๐ท: (โโ, โ) ๐ : (โโ, 6]
Composition and Inverse Functions
1. ๐ฅ2 โ 16
2. โ ๐ฅ โ 12
3. ๐ฅ3 โ 4๐ฅ2 4. ๐ฅ + 3 5. โ12 6. 36
7. ๐โ1(๐ฅ) = ๐ฅ + 4
๐โ1(๐ฅ) = ยฑโ๐ฅ