Piecewise and Rational Functions Review

Piecewise and Rational Functions Review

2021 Final Exam Review

Evaluating Piecewise Functions

• Look at the number that is in the function notation and determine which of the domain restrictions it fits into.

• Plug the x-value into the corresponding equation.

This function is for the next three problems, so write it down now!

Fill in the Blanks

Fill in the Blanks

Fill in the Blanks

Graphing Piecewise Functions (Feuding Neighbors)

• Draw the Fences (domain restriction numbers)

• Determine yard ownership (where the functions belong)

• Determine who owns the yards (whoever has the "equal to" inequality)

• Plot their steps (table of values starting with the fence numbers) **Don't forget about the open/closed circles**

Multiple Choice

Match the graph with the correct equation.

Multiple Choice

Match the graph with the correct equation.

Multiple Choice

Match the graph with the correct equation.

Rational Expressions- Simplifying

• FACTOR EVERYTHING

• Cancel out common factors in the numerator and denominator

Rational Expressions- Multiplying

• All steps from simplifying

• Diagonally cancel any common factors that are in the numerator of one rational expression and the denominator of the other

• Multiply straight across

• Simplify by FOIL or CLT

Rational Expressions-Dividing

• Keep-Change-Flip (turning it into a multiplying problem)

• Same steps as multiplying rational expressions

Multiple Choice

Simplify

• (x-9)/(x+9)

• 1/(x-9)

• (x+9)

• 1/(x+9)

Multiple Choice

Multiply the Rational Expression

• A

• B

• C

• D

Multiple Choice

Divide the Rational Expression

• A

• B

• C

• D

Rational Expressions- Adding and Subtracting

• Factor everything

• Write with the same denominator

• Add/subtract the numerators

• Combine Like Terms

• Factor the numerator again

• Cancel where possible (if nothing can cancel, FOIL again)

Multiple Choice

Multiple Choice

Rational Functions- Graphing

• Domain: set the denominator not equal to zero and solve for x

• Vertical Asymptote: set the denominator equal to zero and solve for x

• Holes: factors that cancel out in the numerator and denominator; set equal to zero and solve

Multiple Choice

What is the domain?

• All real numbers

• All real numbers except 4

• All real numbers except 2

• All real numbers except 2 and -2

Multiple Choice

What is/are the vertical asymptote(s)?

• x=-2
• x=2 and x=3
• x=-3
• x=3

Multiple Choice

What (if any) holes exist?

• x=2

• x=2 and x=3

• x=-3

• x=3

Rational Functions- Horizontal Asympototes

• Find the degree of the numerator (highest exponent or the sum of the multiplicities of each factor)

• HIGHER/LOWER -> there is no HA

• SAME/SAME -> y= ratio of the leading coefficients

• LOWER/HIGHER -> y=0

Multiple Choice

 $h\left(x\right)=\frac{2x^4+8x+6}{3x^3-7x^2}$  

Horizontal asymptote?

•  $y=0$  

•  $y=\frac{2}{3}$  

•  $x=\frac{2}{3}$  

• None

Multiple Choice

Find the horizontal asymptote:

• y = 0

• y = 1

• None

• y = 3/2

Multiple Choice

What is the horizontal asymptote of the function?

• y=0

• y=1/3

• none

Multiple Choice

What are the asymptotes?

• x=1, x= 2, y =1, y= 2
• x= 2, x=-2, y = 1
• x=2 y =-1
• x=1 y =2, y =-2

Multiple Choice

Which graph represents the function?

 $f\left(x\right)=\frac{1}{x-4}+3$