Rational Functions
Answer: 1/3; 1/3
Practice Problem:
Answer: Removable Discontinuity Vertical Asymptote
Practice Problem:
Answer: Vertical Asymptote Removable Discontinuity Zero
Practice Problem:
Answer: Zero Vertical Asymptote Vertical Asymptote
Practice Problem:
Answer: [(x+3)(x+1)]/-10 AC
Practice Problem:
Answer: [(x-4)(x-8)] / [(x-2)(x-1)] ABCD
Practice Problem:
Answer: 2 / (x - 1) A; D
Practice Problem:
Answer: Zero Vertical Asymptote
Practice Problem:
Answer: -4, -4
Practice Problem:
Answer: A
Practice Problem:
Answer: A, D
Practice Problem:
Answer: B
Practice Problem:
Answer: D
Practice Problem:
Answer: (x-4) / -2*(x-3) ACD
Practice Problem:
Answer: (x-4)/(x-3) C
Practice Problem:
Answer: (x-7)/6x, bd
Practice Problem:
Horizontal: What does f(x) equal when x is approaching -infinity and infinity Vertical: What does x equal to make f(x) undefined that is NOT a removable discontinuity.
Difference between horizontal and vertical asymptotes?
Answer: (2y - 3)(2y+5)
Factor: 4y^2 + 4y - 15
What x value is factored out of the num. and den. when f(x) is simplified. Looks like a hole in the graph.
If f(x) equals a rational expression, what does it mean to have a removable discontinuity?
Any x value that makes a function equal zero. Which means any x value that makes the numerator equal zero that is NOT a removable discontinuity.
If f(x) equals a rational function, what does it mean if it has a zero?
What x value causes f(x) to be undefined. Or... what x value makes the denominator equal zero that is NOT a removable discontinuity.
If f(x) equals a rational function, what does it mean to have a vertical asymptote?
Answer: 0; 0
Practice Problem:
YES, all x's that would make the equation undefined must be listed.
If you are dividing rational expressions... do you have to list restricted x's before you multiply by the reciprocal?
Vertical asymptote because simplified function is still undefined at that x value. No need to list it but still can. THIS does not work for zero's, still rem. disc. if multiple of same factor in numerator
If you cancel out two factors of a regular expression in the numerator and denominator, and there is another of that SAME factor in the denominator... is that a removable discontinuity or vertical asymptote?
(x^2 + 3x + 54) / [(x+6)(x-6)(x-6)]
Practice Problem:
List restricted x's (any x value that makes denominator 0) Do this before simplifying. **Note: If you are dividing or multiplying rational equations; any x's in denominators get listed. If dividing; the entire bottom equation x's must be listed**
Reminder, what must you do when simplifying/solving any equation with a polynomial in the denominator?
1. Divide the num. and den. by highest degree x value. 2. Plug in infinity or -infinity to see how the expression would react.
Steps for solving end behavior of rational functions? or... What is f(x) approaching as x is approaching -infinity or infinity? *This is also finding the horizontal asymptote*
1. Find two numbers (a and b) that multiply to the product of the leading and last coefficient. And add up to the middle coefficient. 2. Split the middle coefficient into a and b. 3. Take the first two elements and factor; take the last two elements and factor. 4. Group together into two elements.
Steps of factor by grouping:
1. Factor the numerator and denominator 2. List restricted values 3. Cancel common factors 4. Final answer (simplified expression with list of restricted x values)
Steps to reducing rational expression to lowest terms:
The numerator and denominator have no factors in common.
What does it mean to reduce a rational expression to its lowest terms?
An expression that consists of a sum of terms containing integer powers of x, like 3x^2 - 6x -1.
What is a polynomial?
-Simply a quotient of two polynomials Ex. 1/x ; (x+5) / (x^2 - 4x +4); etc. *Numerator can be a constant and the polynomials can be of varying degrees and in multiple forms*
What is a rational expression?
The set of all possible input values.
What is the domain of any expression?
List of restricted x values in the domain.
If you want to reduce a rational expression to lowest terms, what must you include in the final answer to make the simplified expression equal to the original?
Answer: (-10x^2 + 7x + 56) / [(2x)(x+8)(x+9)]
Practice Problem:
Answer: (-3x-48) / [(x-2)*(x+7)]
Practice Problem:
Answer: (-4x + 12) / (3x - 18) AD
Practice Problem:
Answer: (-54x^2 + 4x + 20) / [(9x)(x+5)(x-5)]
Practice Problem:
Answer: (11x + 23) / [(x+4)(x+4)(x-3)]
Practice Problem:
Answer: (11x-30) / [(x+3)(x-4)]
Practice Problem:
Answer: (2x^2+3x-35) / [(x-5)(x-8)(x-2)]
Practice Problem:
Answer: (4x^2 - 7x + 56) / [(4x)(x+8)(x-8)]
Practice Problem:
Answer: (5x+56) / [(x)(x+8)]
Practice Problem:
Answer: (8x + 55) / [(x-8)(x+9)]
Practice Problem:
Answer: (8x^2 + 29x + 35) / [(x+3)(x+7)(x-7)]
Practice Problem:
Answer: (x+4) / [(x-5)(x-1)] ABCD
Practice Problem:
Answer: (x+6)/(x-6), ac
Practice Problem:
Answer: (x+6)/7 ; C
Practice Problem:
Answer: (x+8) / 20 x = 7, 8
Practice Problem:
Answer: (x-1)/5x ; ab
Practice Problem:
Answer: -infinity; -infinity
Practice Problem:
Answer: -infinity; infinity
Practice Problem:
Answer: 0,0
Practice Problem:
Answer: 1/(x+7) ABD
Practice Problem:
Answer: 3x-15 AC
Practice Problem:
Answer: 4 / (-3x - 18) ABD
Practice Problem:
Answer: B
Practice Problem:
Answer: C
Practice Problem:
Answer: Infinity -infinity
Practice Problem:
Answer: C
Practice Problem:
All real numbers except for those that make its denominator zero.
What is the domain of a rational expression?
Put each in parentheses to properly distribute potential negatives.
When adding and subtracting rational expressions with like denominators... what should you do with the numerators?
Make them like denominators (least common multiple) by factoring (if necessary) and multiplying the other rational expression with the unlike part in the numerator and denominator.
When adding and subtracting rational expressions with unlike denominators... What should you do?
When you are factoring a polynomial with a coefficient greater than one. Example: 4x^2 +4x - 15
When should you use factor by grouping?
Removable discontinuities directly after factoring. Then zero's and vertical asymptote's.
When solving for zero's, vertical asymptote's, and removable discontinuities... what should you solve for first?
