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 Blank

Evaluate: g(-3)

Fill in the Blank

Evaluate: g(0)

Fill in the Blank

Evaluate: g(16)

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?

 $f\left(x\right)=\frac{1}{x-4}+3$