Precalculus Review

​Let's Practice...

​Let's practice...

​The discriminant is under the radical sign.

Multiple Choice

Solve the Following Quadratic Equations

$$x^2-3x-28=0$$  ?

• $$\left\{4,\ -7\right\}$$  

• $$\left\{-2,\ 14\right\}$$  

• $$\left\{-4,\ 7\right\}$$  

• $$\left\{-25,\ 28\right\}$$  

Multiple Choice

Find all the solutions for x when

x² + 9x - 36 = 0

• A.

  (x + 12)(x - 3)

• B.

  x = 12 and x = -3

• C.

  x = -12 and x = 3

• D.

  (x - 9)(x - 4)

Multiple Choice

$$x^2-7x=0$$  

Solve by the method of your choice.

• A.

  x = 7 only

• B.

  x = 0 only

• C.

  x = 0 and 7

• D.

  There is no real answer for this equation.

Multiple Choice

What are the solutions to the equation: $$2x^2+5x+3=0$$  ?

• $$\left\{-3,\ \frac{1}{2}\right\}$$  

• $$\left\{-1,6\right\}$$  

• $$\left\{-\frac{3}{2},\ -1\right\}$$  

• $$\left\{1,\ 5\right\}$$  

Multiple Choice

• x = 12

• x = 24

• x = 144

• x = 121

Multiple Choice

• n = 5 and -6

• n = -6

• n = 5

• No Solution

|x| = 5​

is read as "x is 5 units from 0"​

Absolute Value represents a DISTANCE

EXAMPLE #1: 

EXAMPLE #2: 

SOLVING AN ABSOLUTE VALUE EQUATION

• STEP 1: Get absolute value on one side of equal sign by itself.

• STEP 2: Split into 2 equations (one for positive answer and one for negative)

• STEP 3: Solve each of the equations.

• STEP 4: Write answer in set notation.

• (next slide/video walks you through these steps)

Multiple Choice

∣ 2x + 9 ∣ = 15

• {3}

• { -6,3}

• { -12,3}

• { -12,6}

Multiple Choice

−2|−2r − 4| = -12

• {5,1}

• {-5,1}

• {-5}

• No Solution

Multiple Choice

Solve the inequality.

x + 5 ≤ 13

• x ≥ 8

• x ≥ 18

• x ≤ 8

• x ≤ 18

Multiple Choice

Solve the absolute value inequality.

$$\left|2x-1\right|<3$$  

• $$x<-1\ or\ x>2$$  

• $$1<x<2$$  

• $$-1<x<2$$   

• $$-1>x>2$$  

Multiple Select

Which two inequalities below would be the next step in solving 

$$\left|x+6\right|<2$$  

• $$x+6<-2$$  

• $$x+6>-2$$  

• $$x+6<2$$  

• $$x+6>2$$  

• $$x+6=2$$  

Interval Notation

Interval notations is another way to write an inequalities. It is very useful when describing the graph of inequalities. 

Multiple Choice

Express the following in Interval Notation

-8 ≤ x < 8

• [-8, 8]

• (-8, 8]

• [-8,8)

• (-8, 8)

Multiple Choice

Write the following in interval notation

x < 8

• (8, ∞)

• (-∞, 8)

• (8, -∞)

• [-∞, 8]

Multiple Choice

What would be the next step to isolate the absolute value? $$9\left|x+4\right|\le45$$  

• Divide by 9

• Add 9

• Subtract 4

• Multiply by 9

Multiple Select

Which two inequalities below would be the next step? Click two below

$$\left|x+4\right|\le5$$  

• $$x+4\le-5$$

• $$x+4\ge5$$

• $$x+4\le5$$

• $$x+4\ge-5$$

Multiple Choice

Solve the absolute value inequality.

$$6\left|x+3\right|-10\ge-4$$  

• $$x\le-4\ or\ x\ge-2$$  

• $$-4<x<-2$$  

• $$-2<x<4$$   

• $$x\le-4\ or\ x\ge2$$  

Multiple Choice

Solve the quadratic inequality: $$x^2-14x+48>0$$  

• $$x<6\ ,\ \ x>8$$  

• $$6<x<8$$  

• $$-8<x<-6$$  

• $$x<-8\ ,\ \ x>-6$$  

Multiple Choice

Solve the quadratic inequality: $$x^2-13x+22\le0$$  

• $$x\le-11\ ,\ \ x\ge-2$$  

• $$-11\le x\le-2$$  

• $$2\le x\le11$$  

• $$x\ge11\ ,\ \ x\le2$$  

Multiple Choice

Solve the quadratic inequality: $$x^2+9x+18<0$$  

• $$x<-6\ ,\ \ x>-3$$  

• $$-6<x<-3$$  

• $$x<3\ ,\ \ x>6$$  

• $$3<x<6$$  

How do we solve this problem?​

​

​Correct Answer:

1. ​Combine Like Terms

   Not necessary for this problem

2. Factor out polynomials​

​

​

​

​

​

3. Cancel Out Common Factors​

   We can cancel ou​t (x-2), you will then have the correct answer

​

​

​

​

​(x-6)(x-2)

How do we solve this problem?​

​

​Correct Answer:

1. ​Combine Like Terms

   This is not necessary for this problem

2. Factor Polynomials​

​

​

​

​

3. Cancel out Common Factors​

We can cancel out (n+5), giving us the correct answer.​

​

​

Multiple Choice

Simplify the Expression

Multiple Choice

Simplify the Expression

Multiple Choice

Simplify

$$\frac{12x^2-54x}{6x}\cdot\frac{6}{81-18x}$$  #7

• $$-\frac{2}{3}$$  

• $$-1$$  

• $$\frac{x-7}{5x^2}$$  

• $$\frac{x-4}{6}$$  

Multiple Choice

Divide

$$\frac{x^2+10x+21}{x^2-9}\div\frac{x+7}{x-7}$$  #15

• $$\frac{x-10}{x-3}$$  

• $$\frac{10x+7}{3}$$  

• $$\frac{x+3}{x-7}$$  

• $$\frac{x-7}{x-3}$$  

Multiple Choice

Divide the expression  $$\frac{x+8}{6\left(x+10\right)}\div\frac{\left(x+8\right)\left(x-8\right)}{\left(x+10\right)\left(x-8\right)}$$  #14

• $$\frac{1}{6}$$  

• $$\frac{\left(x+10\right)}{6x+60}$$  

• $$\frac{x+8}{6x+60}$$  

• $$\frac{\left(x+10\right)\left(x+8\right)}{6x+60}$$  

Multiple Choice

The domain of the rational expression is: $$\frac{\left(x-11\right)\left(x+12\right)}{\left(x-10\right)\left(x+2\right)}$$  #2

• all real numbers except: -10, 2

• all real numbers except: -11, 12

• cannot be found

• all real numbers except: 10, -2

Solving rational equations

• 1. Factor the DENOMINATORS only

• 2. Multiply each side of the equation by the least common multiple (LCM) this should get rid of the fraction

• 3. Solve

• 4. Check to make sure your answers work (you don't divide by zero)

Multiple Choice

Solve for x.

• -4

• 0

• $$\frac{1}{2}$$

• 4

Multiple Choice

Try on your own. Solve for y.

• -1

• 0

• 1

• -2

Check your answers 

Multiple Choice

Solve the rational equation.

$$\frac{5}{x}+\frac{8}{3}=1$$  

• $$x=0$$  

• $$x=3$$  

• $$x=9$$  

• $$x=-9$$  

Multiple Select

Solve

$$\frac{x+1}{x-3}=-\frac{x^2-6x+1}{\left(x-3\right)\left(x-1\right)}$$  

• -5

• -2

• 0

• 1

• 4

Multiple Choice

Solve the rational equation

$$\frac{2x}{x+1}-\frac{3}{2}=\frac{x}{x+1}$$  

• $$x=-3$$  

• $$x=3$$  

• $$x=-1$$  

• $$x=-\frac{3}{2}$$  

Multiple Choice

Solve the rational equation

$$\frac{x}{x-2}+\frac{1}{x-4}=\frac{2}{x^2-6x+8}$$  

• $$x=4,\ x=-1$$  

• $$x=-3,\ x=-4$$  

• $$x=-1$$  

• $$x=-4,\ x=1$$  

Function: Relates the input to output (connects them together).

Input: X-values that go into the function.

Output: The result you get from plugging in the x-values.​

Multiple Choice

If f(x) = 2x + 1, find f(1).

• f(1) = 1

• f(1) = 4

• f(1) = 0

• f(1) = 3

Fill in the Blank

What is a composite function?

• Can be thought of a series of operations

• One function is substituted into another function

• Often denoted as f(g(x)) or (f ∘ g)(x)

• Read as "f of g of x"

Example

Multiple Choice

Identify if the following is a composite function: $$f\left(x\right)\cdot g\left(x\right)$$  

• Yes

• No

• It is an operation on functions

• You are multiplying f(x) and g(x) not substituting functions

Multiple Choice

Which expression represents g(f(x)) if

f(x) = 2x - 8 and

g(x) = 4x

• 8x - 32

• 8x² - 32x

• 8x - 8

• 6x - 8

Multiple Choice

What is f(g(3))?

• 1/8

• 8

• 1/4

• 1/2

Multiple Choice

Given f(x)= -3x+7 and g(x)=2x² - 8, find g(f(x)).

• g(f(x))= -6x²+31
• g(f(x))= -6x²+24
• g(f(x))=18x²-84x+90
• g(f(x))=9x²-42x-41

Multiple Choice

f(x) = 3x + 10
g(x) = x - 2
Find (f∘g)(0)

• 16
• 4
• -4
• None of these.

Multiple Choice

State if the given functions are inverses.

Determine this by seeing if $$fog\left(x\right)\ =\ x$$  

and $$gof\left(x\right)=x$$  .

• yes

• no

Multiple Choice

Find the inverse of f(x) = -4x - 12. #6

• f^-1(x) = 4x - 3
• f^-1(x) = -1/4x - 3
• f^-1(x) = 1/4x + 3
• f^-1(x) = -4x - 3

Multiple Choice

Find the inverse of

$$f\left(x\right)\ =\ \sqrt{x-5}$$  . #21

• $$f^{-1}\left(x\right)\ =\ x^2-\ 5$$  

• $$f^{-1}\left(x\right)\ =\ x^2+5$$  

• $$f^{-1}\left(x\right)\ =\ \left(x+5\right)^2$$  

• $$f^{-1}\left(x\right)\ =\ \left(x\ -5\right)^2$$  

Logs are the inverse of Exponential Functions--their graphs are inverted

2^x is in red

log₂​ x is in blue

Notice the asymptotes?​

​

​What else has switched?

​

Graphs of Log Functions

We will focus on what a, h, and k do to affect the parent function

​

​Transformations of Logs

The parent graph, log₂​(x)

The transformed graph,

log₂​(x+3)

Which way did the graph move? Where is the asymptote?​

​what is the h-value for the red graph?

The asymptote, h

​

Multiple Choice

Identify the asymptote of the transformed graph $$\log_2\left(x+3\right)$$  shown in red in the graph

• $$x=-3$$  

• $$x=3$$  

• $$y=3$$  

• $$y=-3$$  

Multiple Choice

Which of the equations could best describe the graph?

• $$\log_2\left(x+1\right)$$  

• $$\log_2\left(x-1\right)$$  

• $$\log_2\left(x\right)+1$$  

• $$\log_2\left(x\right)-1$$  

Multiple Choice

Which graph correctly shows the asymptote of $$f(x)=\log(x+2)+3$$  

Match

Match the following graphs with their domains

Domian: x > -2

Domian: x > 3

Domian: x > 0

Domian: x > 2

• Domian: x > -2

• Domian: x > 3

• Domian: x > 0

• Domian: x > 2

Fill in the Blank

$$\log_525=$$  

Fill in the Blank

$$\log_381=$$  

Fill in the Blank

$$\log_{36}6=$$  

Fill in the Blank

$$\log_9\frac{1}{9}=$$  

Fill in the Blank

$$\ln e^4=$$  

Fill in the Blank

$$\ln\sqrt{e}=$$  

Fill in the Blank

$$\log_81=$$  

Multiple Choice

Rewrite 1.6^(x+2) = 14.1

• log_1.614.1 = x+2

• log_1.6(x+2) = 14.1

• log_14.1(x+2) = 1.6

• log_(x+2)1.6 = 14.1

Multiple Choice

Rewrite log_x95 = 2.3

• x^2.3 = 95

• 95^2.3 = x

• 2.3⁹⁵ = x

• x⁹⁵ = 2.3

Multiple Choice

$$\log\left(\frac{m^3}{n}\right)$$  

• logm-log3-logn

• 3logm+logn

• 3logm-logn

• 3log(m-n)

Multiple Choice

$$\log\left(\frac{8x}{yz}\right)\$$  

• log8+logx-logy-logz

• log8+logx-logy+logz

• log(8x)-log(yz)

• 8logx-ylogz

Multiple Choice

• log (xy³)

• log (x⁶ − y³)

• log (x⁶/y³)

• log (x⁶ + y³)

Multiple Choice

• ln(wvu⁵)

• ln(u + v + w⁵)

• ln(uvw⁵)

• ln(uvw)⁵

Multiple Choice

Solve the following for 'x';

log₆(3x - 2) = log₆(5x - 8)

• x = 3

• x = 2

• x = -1

• No Solution

Multiple Choice

log₂(x+3)=4

• 16

• 13

• 3

• 10

Multiple Choice

• 5

• 13

• 84

• -20

Multiple Choice

Solve

• 46/5

• 5/46

• -46/5

• -5/46

Multiple Choice

log(x+6) = 1

• 4

• 4.222

• -5

• -9.550

Multiple Choice

• -.20

• .60

• .20

• -.6

Multiple Choice

Solve for x.

e^x+6 + 5 = 1

• -4.614

• -4.236

• No Solution

• -0.334

Multiple Choice

Solve for x: log₆x + log₆(x-5) = 2

• √7

• 9

• 4

• none of these

Multiple Choice

State the possible rational roots of the function.

Multiple Choice

State the possible rational zeros.

Multiple Choice

State all of the possible rational roots.

Multiple Choice

$$\frac{10x^3+37x^2+37x+6}{5x+1}=?$$  

• $$2x^2+4x+6$$  

• $$2x^2-7x-6$$  

• $$2x^2-7x+6$$  

• $$2x^2+7x+6$$  

Multiple Choice

What is the final answer?

$$\frac{3x^3-4x^2-9x+10}{x-2}=?$$  

• $$3x^2+2x-5$$  

• $$3x^3+2x^2-5x+0$$  

Multiple Choice

Divide

$$\left(n^3+8n^2-2n+63\right)\div\left(n+9\right)$$  

• $$n^2-n+7$$  

• $$n^2+n-7$$  

• $$n^2-n-7$$  

• $$n+7$$  

Multiple Choice

The function shown has:

• Degree two with negative leading coefficient.

• Degree three with negative leading coefficient.

• Degree four with negative leading coefficient.

• Degree four with positive leading coefficient.

Multiple Select

Select all the graphs which show even degree functions. #2

Multiple Choice

#5

• y =  ^_(x + 2)(x - 1)²
• y = -(x - 2)(x + 1)²
• y = (x - 2)(x + 1)²
• y = -x³

Multiple Choice

What are the zeros of the function? #9

• 4,1
• 3,1,-2,-5
• -3, -1, 2, 5
• -2,4

Multiple Choice

Find all zeros for the polynomial function P(x) = x³ -3x²-5x+15. #23

• 3, √5 
• 3, √5, -√5
• -3, √5, -√5
• -3, 5i, -5i

Multiple Choice

Factor:  x³-12x²+47x-60=0 when 5 is a root.

• (x-3)(x-4)(x-5)=0
• (x+1)(x-4)(x-5)=0
• (x-3)(x-4)(2x-5)=0
• (x-5)(x-4)(x-5)=0

Multiple Select

Given  $$x=-3$$  is a zero of the polynomial  $$f\left(x\right)\ =\ x^3-13x-12$$  , use synthetic division to find all of the other zeros.  Choose ALL that apply.

• $$x=1$$  

• $$x=4$$  

• $$x=-1$$  

• $$x=-4$$  

• $$x=3$$  

​complex number

​A number in the complex plane. It can be written as a + bi, where a and b are real numbers and i² = -1.

Open Ended

Simplify the expression:

(3 + 9i)(5i)

Multiple Choice

#5

• 10
• -10
• 10i
• -10i

Multiple Choice

• A
• B
• C
• D

Multiple Choice

Solve
3x² - 5 = -446

• ±√-147
• 7i√3
• ±7i√3
• -7√3

Just combine like terms! The "real" parts, the -2 and 1, combine to make -1 and the "complex" parts 5i and -7i, combine to get -2i.

​

Simplify

(-2 + 5i) + (1 - 7i)​

Multiply the same way you would basic binomials! There is just ONE extra step...

​Multiplying Complex Numbers

​

Multiple Choice

$$Simplify:\ \ \left(-2\ +\ 7i\right)\left(4\ -\ 2i\right)$$  

• $$6+\ 32i$$  

• $$-6\ +\ 32i$$  

• $$22+32i$$  

• $$6\ +\ 24i$$  

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Multiple Choice

Add $$\frac{2x-3}{4x-7}+\frac{2x-3}{4x-7}$$  #25

• $$\frac{4x-6}{4x-7}\ ,\ x\ne-\frac{7}{4}$$  

• $$\frac{4x^2-6x+9}{4x-7}\ ,\ \ x\ne\frac{7}{4}$$  

• $$\frac{4x-6}{4x-7\ }\ ,\ x\ne\frac{7}{4}$$

• $$\frac{4x-6}{8x-14},\ x\ne\frac{14}{8}$$  

Multiple Choice

Simplify the expression. #29

• n / 2m

• n / 3m

• (m² + 4n²) / 12m

• n / 4m