Find the domain:x=-7, x=3

Problem 1:

Step 1: Find GCF

8(x5-x4-42x3+106x2+41x-105)

8x5-8x4-336x3+848x2+328x-840 8

Step 2: Divide by given “x” values

x5-x4-42x3+106x2+41x-105x+7x4-8x3+14x2+8x-15

-(x5+7x4)-8x4-42x3

-(-8x4-56x3)14x3+106x2

-(14x3+98x2)8x2+41x

-(8x2+56x)-15x-105

-(-15x-105)0

x4-8x3+14x2+8x-15x-3x3-5x2-x+5

-(x4-3x3)-5x3+14x2

-(-5x3+15x2)-x2+8x

-(-x2+3x)5x-155x-15

0

1 2

What we have so far:

8(x+7)(x-3)(x3-5x2-x+5)

Let’s reduce this.

Step 3: Factor by Grouping

x3-5x2-x+5

(x3-5x2)-1(x-5)

(x2-1)(x-5)

Step 4: Difference of Squares

x2-1

(x+1)(x-1)

Put it all together…

(x-3)(x+7)(x+1)(x-1)(x-5)

…under the radical.

√( x −3)( x+7)( x+1)( x−1)( x−5) 

…and find where it’s positive (or zero)!

D: [-7,-1]U[1,3]U[5,∞)

Problem 2:Solve for “z” given the following information:

•7x=343•xx-4x2+15x-7=y•logy((20-z)(z+3)+1)=2

Step 1:

Solve for “x” by converting 7x=343 to a log.

log7 (343)=x

x=3

*The purpose of logs is to determine the value of an exponent that is shown as a variable in an exponential function. Logs make it possible to figure out this “x” value based on the original base number and the “y” value.

Step 2:

Plug 3 (x-value) into “xx-4x2+15x-7=y”

33−4 (3 )2+15 (3 )−7=𝑦

9-36+45-7=y

Y=11

Step 3:

Plug 11 (y-value) into “logy((20-z)(z+3)+1)=2”

log11((20-z)(z+3)+1)=2

112 = (20-z)(z+3)+1

120 = -z2+17z+60

60 = -z2+17z

Now graph, and find wherey = 60

The Table

And there you go.

Z=5

Calculate:• Hole(s)• X-intercept(s)• Y-intercept• Horizontal Asymptote• Vertical Asymptote(s)

and graph.

Problem 3:__4x3-28x2-4x+28__(x2-3x-4)(2x2-22x-56)

Step 1:Factor the numerator and denominator.

__4x3-28x2-4x+28__(x2-3x-4)(2x2-22x+56)

Factor by GroupingFactor Normally

Step 1a:

4x3-28x2-4x+28 (4x3-28x2)-1(4x-28)

(4x3-28x2)-1(4x-28) 4x2(x-7)-4(x-7)

4x2(x-7)-4(x-7) (4x2-4)(x-7)

(4x2-4)(x-7) (2x+2)(2x-2)(x-7)

Our new numerator:

_(2x+2)(2x-2)(x-7)_(x2-3x-4)(2x2-22x+56)

Now, for the denominator…

Step 1b:

(x2-3x-4)(2x2-22x+56) (x-4)(x+1) (2x2-22x+56)

(x-4)(x+1) (2x2-22x+56) 2(x-4)(x+1)(x2-11x+28)

2(x-4)(x+1)(x2-11x+28) 2(x-4)(x+1)(x-7)(x-4)

Our Fully Factored Fraction (FFF)

_(2x+2)(2x-2)(x-7)_2(x-4)(x+1)(x-7)(x-4)

Calculations:Holes can be found where the same factor is

found in both the numerator and the denominator.

_(2x+2)(2x-2)(x-7)_2(x-4)(x+1)(x-7)(x-4)

Hole at x=7

Calculate:• Hole(s): x=7• X-intercept(s)• Y-intercept• Horizontal Asymptote• Vertical Asymptote(s)

Calculations:X-intercepts can be found where the

numerator is 0.

(2(-1)+2)(2(1)-2)((7)-7)2(x-4)(x+1)(x-7)(x-4)

Calculate:• Hole(s): x=7• X-intercept(s): x=-1, 1• Y-intercept• Horizontal Asymptote• Vertical Asymptote(s)

x=-1 x=1DO NOT INCLUDE THISBecause it’s already a Hole.

Calculations:Y-intercepts can be found when you plug in 0.

_(2(0)+2)(2(0)-2)(0-7)_2(0-4)(0+1)(0-7)(0-4)

__(2)(-2)(-7)__2(-4)(1)(-7)(-4)

-224Calculate:• Hole(s): x=7• X-intercept(s): x=-1, 1• Y-intercept: y= -1/8

• Horizontal Asymptote• Vertical Asymptote(s)

28

Calculations:Asymptotes:

Horizontal: -If numerator is lower power, it’s just 0. -If numerator and denominator are the

same powers, divide leading coefficient.Vertical: -present when

denominator is 0.

Calculate:• Hole(s): x=7• X-intercept(s): x=-1, 1• Y-intercept: y= -1/8

• Horizontal Asymptote: 0• Vertical Asymptote(s)

_(2x+2)(2x-2)(x-7)_2(x-4)(x+1)(x-7)(x-4)

Power=3Power=4

Calculations:Asymptotes:

Horizontal: -If numerator is lower power, it’s just 0. -If numerator and denominator are the

same powers, divide leading coefficient.Vertical: -present when

denominator is 0.

Calculate:• Hole(s): x=7• X-intercept(s): x=-1, 1• Y-intercept: y= -1/8

• Horizontal Asymptote: 0• Vertical Asymptote(s): x=-1, 4

_(2x+2)(2x-2)(x-7)_2((4)-4)((-1)+1)((7)-7)((4)-4)

DO NOT INCLUDE THISBecause it’s already a Hole.

Final Information

Hole(s): x=7X-intercept(s): x=-1, 1

Y-intercept: y= -1/8

Horizontal Asymptote: 0Vertical Asymptote(s): x=-1, 4

Because….

Sign Chart

X

Y

_(2x+2)(2x-2)(x-7)_2(x-4)(x+1)(x-7)(x-4)

-1- -1+ 4- 4+

_( - )( - )( - )_( - )( - )( - )( - )

_( + )( - )( - )_( - )( + )( - )( - )

_( + )( + )( - )_( - )( + )( - )( - )

_( + )( + )( - )_( + )( + )( - )( + )

- - + +

Estimated graph:

End of Jacob’s Slides

4. Find f(g(x)), simplify and draw the graph

F(x)= 4x + 3

G(x)= 2x-1

Step 1) Replace the x variable with 2

x-1

This will give you

4 + 32x-1

Now, consolidate by multiplying the numerator by 4 to get

+ 38x-1

Beginning of Vlad’s slides

Find f(g(x)), simplify and draw the graph

The next step is to find a common denominator. In order to do this, multiply the 3 by the quantity (x-1) to get 8

x-13x-3x-1

+

Now you combine the terms and you are left over with

3x+5x-1

X-intercepts are found when the numerator = 0. The x-intercept is -5/3.

Y-intercept is are found when you plug in 0 into the equation The y-intercept is -5

Find f(g(x)), simplify and draw the graph

x-int = -5/3Y-int = -5

3x+5x-1

The vertical asymptote is where the denominator = 0 because you cannot divide by 0. The VA is -1

The Horizontal asymptote is 3 because the numerator and denominator are to the same power, so you divide the leading coefficients to get 3.

Put it all together and you get …….

-5/3

3

-5 Assume all lines go to infinity

5. Simplify, find any holes, x intercepts, asymptotes, and graph the function.

x3 + 25x x2 -x -20(x-2)

Upon looking at the numerator, you can see that it can be solved by using a difference of squares after you factor out an x first.

The difference of squares is (a+b)(a-b)

The first step is to simplify the numerator and denominator as much as possible

x(x+5)(x-5)x2 -x -20(x-2)

Now, you factor out the denominator. You should now have

x(x+5)(x-5)(x+4)(x-5)(x-2)

You can now find your x-intercepts, holes, and graph the function

You should now have

Simplify, find any holes, x intercepts, asymptotes, and graph the function.

x3 + 25x x2 -x -20(x-2)

x(x+5)(x-5)(x+4)(x-5)(x-2)

Your x intercepts is where you numerator equals 0. You can get 0 in the numerator by plugging in 0 and-5, so these numbers would be your x intercepts. 5 would not be an x-intercept because it is a hole.

-5 5Your y-intercept occurs when you plug in 0 for you x values. Y-int: 0

Holes occur when your numerator and denominator equal 0. The hole for this graph would be 5 because the numerator and denominator are both multiplied by the quantity (x-5).

Simplify, find any holes, x intercepts, asymptotes, and graph the function.

x(x+5)(x-5)(x+4)(x-5)(x-2)

The horizontal asymptote for this function would be 1 because the numerator and denominator are to the same power and have the same leading coefficient.

The vertical asymptotes' of this function would be -4 and 2 because the vertical asymptote is found when the denominator equals 0

-5 5-4 2

1

(-)(+)(-)(-)(-)(-)

x

y

-4 -4 2 2Step 3) Graph it using a sign chart

Simplify, find any holes, x intercepts, asymptotes, and graph the function.

-5 5-4 2

1

x(x+5)(x-5)(x+4)(x-5)(x-2)

(-)(+)(-)(+)(-)(-)

-(+)(+)(-)(+)(-)(-)

-(+)(+)(-)(+)(-)(+)

- -+ +

3. Goal: Find the Domain of (x+2)(x-4)(x+5)(x-1)

Step 1) In order to find the domain, we must first find the asymptotes.

The horizontal asymptote for this function would be 1 because the numerator and denominator are to the same power and have the same leading coefficient.

The vertical asymptotes' of this function would be -5 and 1 because the vertical asymptote is found when the denominator equals 0

-5 1

1

Goal: Find the Domain of (x+2)(x-4)(x+5)(x-1)

Step 2) locate x intercepts

X intercepts occur when the numerator equals 0. In this case, the x intercepts

are -2 and 4.

-5 1

1

-2 4

Goal: Find the Domain of (x+2)(x-4)(x+5)(x-1)

Step 3) Graph it using a sign chart

(-)(-)(-)(-)

x

y

-5 -5 1 1

-

(-)(-)(+)(-)

(+)(-)(+)(-)

(+)(-)(+)(+)

-

1

-2 4

Assume all lines go to infinity

Note: you can do it without graphing, but graphing it is easier.

Goal: Find the Domain of (x+2)(x-4)(x+5)(x-1)

The domain of any radical function is = to or greater than 0 on the y axis because you cannot square root a negative number and get a real solution.

1

-2 4

Assume all lines go to infinity

The Domain Would be:

-( , 5) U -2, 1) U 4, )