B: Matrices and Rational Functions

Greg plans to purchase a bicycle. Bike A costs $270 and has an expected maintenance cost of $36 per year. Bike B costs $360 and has an expected maintenance cost of $20 per year. Which statement is true based on the average annual cost of each bike?

Bike A is the better choice if Greg plans on owning the bike for 5 years. Bike B is the better choice if Greg plans on owning the bike for 6 years.

The function y = 16,500+1020x/x models y, the average annual cost in dollars of owning a car for x years. Which statement best explains why the graph of y = 16,500+1020x/x should have a vertical asymptote at x = 0?

The value of the car will never dip below $0. _The time the car is owned is greater than 0. The initial cost of the car is greater than $0.

Where is the removable discontinuity of f(x) located?▫️

x = -2

Determine the equations of the vertical and horizontal asymptotes, if any, for f(x) = 2x/x+4.

x = -4, y = 2

Which solution to the equation 1/x-1 = x-2/x²-2 is extraneous?

x = 1

What is the difference? x/x²+3x+2 - 1/(x+2)(x+1)

x-1/x²+3x+2

What is the difference? x/(x² - 2x - 15) - 4/(x² + 2x - 35)

x²+3x-12/(x+3)(x-5)(x+7)

Janet wants to solve the equation y + y²-5/y²-1 = y²+y+2/y+1. What should she multiply both sides of the equation by?

y _y2 - 1 y + 1 y2 + y + 2

A pharmacist has 80 milliliters of a 25% saline solution. Which expression can be used to find the percentage concentration of the final solution if she adds x milliliters of a 10% saline solution?▫️

y = 0.25(80)+0.1x/80+x • 100

Which statement decribes the behavior of the function f(x) = 3x/4-x?

The graph approaches -3 as x approaches infinity. The graph approaches 3 as x approaches infinity. _The graph approaches 4 as x approaches infinity.

Given x varies inversely with y and xy = 2, what is the value of x when y = 1?

2

What is the quotient? a - 3/7 / 3 - a/21

-(a - 3)²/147 (a - 3)²/147 _3 -3

x/x²+3x+2 + 3/x+1 The numerator of the simplified sum is [_________].

-4x + 2

What is the quotient? 2m + 4/8 / m + 2/6

24/(m + 2)² (m + 2)²/24 2/3 _3/2

Which expression is equivalent to 2a + 1/10a - 5 / 10a/4a² - 1?

2a/(2a - 1)² 50a/(2a + 1)² (2a - 1)²/2a (2a + 1)²/50a

Which expression is equivalent to the following complex fraction? (2 - 1/y)/(3 + 1/y)

2y-1/3y+1

Which expression is equivalent to 3x/x + 1 divided by x + 1?

3x/x + 1 • 1/x + 1 _3x/x + 1 / 1/x + 1 x + 1/1 / 3x/x + 1 x + 1/3x • x + 1/1

What is the common denominator of y + (y - 3/3) in the complex fraction y + (y - 3/3) / 5/9 + 2/3y?

3y(y-3) 3y 3

Which expression is equivalent to the following complex fraction? (2 - 1/y) / (3 + 1/y)

3y+1/2y-1 (2y-1)(3y+1)/y² y²/(2y-1)(3y+1) _2y-1/3y+1

What is the product? 5k/6 • 3/2k³

5/4k²

For the inverse variation equation xy = k, what is the constant of variation, k, when x = -3 and y = -2?

6

What is the sum? 2/x² + 4/x²

6/x²

Which statement decribes the behavior of the function f(x) = 2x/1 - x²?▫️

The graph approaches -2 as x approaches infinity. The graph approaches 0 as x approaches infinity. _The graph approaches 1 as x approaches infinity. The graph approaches 2 as x approaches infinity.

A pharmacist has 80 milliliters of a 30% saline solution. The graph shows the percentage concentration by adding x milliliters of a 10% saline solution. Which statement is true about the graph of the function?

The vertical asymptote x = 30 means the concentration will always be less than 30%. The vertical asymptote x = 10 means the volume will always be greater than 10. _The y-intercept means the concentration will remain 30% if no solution is added. The y-intercept means the concentration will always be less than 30%.

Working alone, it would take Shandra 45 minutes to mow the lawn. To mow the same lawn, also working alone, it would take Kelly 1 hour and 15 minutes. If they work together, the girls believe it will take them 1 hour to mow the lawn. Which best describes the reasonableness of their solution?

Their estimate is longer than the time it will actually take: 1/0.75 + 1/1.25 = 1/time to complete together

What are the vertical and horizontal asymptotes for the function f(x) = 3x²/x²-4?

horizontal asymptote: y = 3 vertical asymptote: x = -2, x = 2

Find the vertical, horizontal, and slant asymptotes, if any, for f(x) = 2x³+24x²+35x-175/x²+14x+45

horizontal: y = 0 slant: y = 2x - 4

What is the vertical asymptote of the function? (f(x) = 3/x+5)

x = -5

What are the equations of the asymptotes of the graph of the function f(x) = 3x²-2x-1/x²+3x-10?

x = -5, x = 2 and y = 3

What are the removable discontinuities of the following function?

x = -6 and x = 6

What are the solutions to the equation x - 7/x = 6?▫️

x = -7 and x = 1 x = -6 and x = -1 _x = -1 and x = 7 x = 1 and x = 6

What is the common denominator of 5/x²-4 - 2/x+2 in the complex fraction (2/x-2 - 3/x²-4)/(5/x²-4 - 2/x+2)?

(x + 2)(x - 2)

What is the difference? (2x + 5)/(x² - 3x) - (3x + 5)/(x³ - 9x) - (x + 1)/(x² - 9)

(x + 5)(x + 2)/x³ - 9x

Which expression is equivalent to the following complex fraction? (3/x - 1) - 4 / 2 - (2/x - 1)▫️

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

(2n/6n + 4)(3n + 2/3n - 2) What is the numerator of the simplified expression? [____________]

-n

Which graph represents the function f(x) = 2/x-1 + 4?

. . . . . . . . . . . . .|. . . | . . . . . . . . . . . . .|. . . | ___________________|_ . . .\_____________ ___________________|__\_________________ . . . . . . . . . . . . .| . .| . . . . . . . . . . . . .| . .| . . . . . . . . . . . . .| . .| . . . . . . . . . . . . .| . .|

Which graph represents the function f(x) = 1/x - 1?

. . . . . . . . . . . . .|| . . . . . . . . . . . . .| | . . . . . . . . . . . . .| | ___________________|__\________________ _______________. . .| . . \______________ . . . . . . . . . . .\ .| . . . . . . . . . . . . . .|| . . . . . . . . . . . . . .|| . .

How many extraneous solutions does the equation below have? 9/n²+1 = n+3/4

0 _1 2 3

What is the product? a - 3/15a • 5/a - 3

1/3a

For the inverse variation equation p = 8/V, what is the value of V when p = 1/2?

16

Which expression is equivalent to 4f²/3 / 1/4f?

16f³/3 f/3 3/16f³ 3/f

A 22,800-gallon swimming pool is being filled by four hoses. Water flows through each hose at a rate of 300 gallons per hour. How many hours will it take to fill the pool?

19 hours

For the inverse variation equation p = 8/V, what is the value of V when p = 4?

2

Which expression is equivalent to the following complex fraction? (2/x - 4/y) / (-5/y + 3/x)

2(y - 2x) / 3y - 5x

n + 3/2n - 6 / n + 3/3n - 9▫️

2/3 _3/2 (n + 3)²/6(n - 3)² 6(n - 3)²/(n + 3)²

What is the quotient? 2y² - 6y - 20/4y + 12 / y² + 5y + 6/3y² + 18y + 27

2/3(y - 5) 3(y - 5)/2 _(y - 5)(y + 2)²/6(y + 3)² 3(y + 5)(y - 2)/2(y + 2)

What is the product? 4k + 2/k^2

2/k + 2

What is the sum? (1/x + 2) + (1/x + 3) + (1/x² + 5x + 6)

2/x+2

What is the sum? 3/x + 4/x²

3x+4/x²

His solution is as follows: Step 1: 8(x - 4) = 2(x + 2) Step 2: 4(x - 4) = (x + 2) Step 3: 4x - 16 = x + 2 Step 4: 3x = 18 Step 5: x = 6 He determines that 6 is an extraneous solution because the difference of the numerators is 6, so the 6s cancel to 0. Which best describes the reasonableness of the student's solution?

His solution for x is incorrect. When solved correctly, there are no extraneous solutions.

What causes a solution to a rational equation to be an extraneous solution?

If a solution results in zero when substituted into the denominator of the equation, the solution is extraneous.

The function f(x) = 30+0.25x/x yields the average cost in dollars per cup of lemonade made by Lincoln at his lemonade stand when he makes x cups of lemonade. Which statement best fits the situation modeled by the function?

Lincoln sells lemonade for 25 cents a cup. Lincoln gave away 30 cups of lemonade for free. _Lincoln uses 25 cents worth of supplies per cup. Lincoln uses 30 cents worth of supplies per cup.

Peter wants to borrow $3,000. He has two payment plans to choose from. Plan A is 4% interest over 6 years. Plan B is 5% interest over 4 years. Using the formula m = P+Prt/12t for payment, m, which statement best compares the plans?

Plan A has a monthly payment of about $23 less and a total interest charge of $120 less than plan B. Plan A has a monthly payment of about $23 less and a total interest charge of $120 more than plan B. _Plan A has a monthly payment of about $23 more and a total interest charge of $120 more than plan B. Plan A has a monthly payment of about $23 more and a total interest charge of $120 less than plan B.

The slant asymptote of f(x) = x²-4x+1/2x-3 is y = 1/2 - 5/4.

T

The function f(x) = 80,000+0.2x/x yields the average cost in dollars for a company to produce x copies of a comic book. Which statement best fits the situation modeled by the function?

The company earns 20% of the issue price as profit from each copy sold. The company earns 20 cents as profit from each copy sold. The company spends 20% of the issue price on materials to produce each copy. _The company spends 20 cents on materials to produce each copy.

How does the range of f(x) = 6/x compare with the range of the parent function f(x) = 1/x?

The range of both f(x) and g(x) is all nonzero real numbers

Which expression is equivalent to the expression below? x/x + 4/x

_x/x + 4 • 1/x x + 4/x • 1/x x + 4/x • x/1

What is the common denominator of 1/a + 1/b in the complex fraction (1/a - 1/b) / (1/a + 1/b)?

ab

What is the domain of f(x) = 3x/x-1?

all real numbers except 1

What is the domain of the function f(x) = x+1/x²-6x+8?

all real numbers except 2 and 4

What is the solution for the equation 5/3b³-2b²-5 = 2/b³-2?

b = 0 and b = 4

Which expression is equivalent to c² - 4/c + 3 / c + 2/3(c² - 9)?

c + 3/c² - 4 • c + 2/3(c² - 9) c² - 4/c + 3 / 3(c² - 9)/c + 2 c² - 4/c + 3 • 3(c² - 9)/c + 2 c² - 4/c + 3 • c + 2/3(c² - 9)

What is the solution to the equation -3d/d²-2d-8 + 3/d-4 = -2/d+2?

d = -4 and d = 2 d = -2 and d = 4 _d = 1 d = 2

Which solution to the equation 3/2g+8 = g+2/g²-16 is extraneous?

g = -4

What is the solution to the equation 1/h-5 + 2/h+5 = 16/h²-25?

h = 11/3 _h = 5 h = 7 h = 21/2

What are the vertical and horizontal asymptotes of f(x) = 2x/x-1?

horizontal asymptote at y = 2, vertical asymptote at x = 1

Kent multiplies both sides of the equation below by an expression. k + 12/k = 8 Then he moves all the terms to one side of the equal sign in the resulting equation. Which equation must he solve now?

k² - 8k - 12 = 0 _k² + 8k + 12 = 0 k² - 8 + 12 = 0 k² - 8k + 12 = 0

What is the solution to the equation 3/m+3 - m/3-m = m²+9/m²-9?

no solution

The function y = 60/x relates y, the time in hours that it takes Michael to travel to his grandparents' house, to x, his average speed in miles per hour. What does the number 60 in the function represent?

the time, in hours, that it takes to get to Michael's grandparents' house _the distance, in miles, that must be traveled to get to Michael's grandparents' house the fuel, in gallons, that are used to get to Michael's grandparents' house the rate, in miles per hour, that Michael travels to get to his grandparents' house

Find the vertical, horizontal, and slant asymptotes, if any, for f(x) = 5x³-51x²+77x+100/x²-11x+24▫️

vertical: x = 8, x = 3 slant: y = 5x + 4

What are the solutions to the equation w/2w-3 = 4/w?

w = 2 and w = 6

Where are the asymptotes for the following function located?

x = -1 and x = 5

Determine the equations of the vertical and horizontal asymptotes, if any, for g(x) = x-2/x²+4x+3.

x = -1, x = -3, y = 0

What is the vertical asymptote of the function? (f(x) = 10/x+2)

x = -2

Determine the equations of the vertical and horizontal asymptotes, if any, for h(x) = (x+1)²/x²-1.

x = 1, y = 1

Which of the following represents the solution of 3/2 = 3/2x - 6/5x?

x = 1/5

Determine the equations of the vertical and horizontal asymptotes, if any, for g(x) = x³/(x-2)(x+1).

x = 2, x = -1

Determine the equations of the vertical and horizontal asymptotes, if any, for f(x) = x/x-5.

x = 5, y = 1

What is true about the solution of x²/2x-6 = 9/6x-18?

x = ±√3 and they are actual solutions.

What is the sum? (x/x + 3) + (3/x + 3) + (2/x + 3)

x+5/x+3

Which expression is equivalent to the following complex fraction? (1 - 1/x)/2

x-1/2x

In a rational function, if there is a vertical asymptote at an x-value, then that value of x will produce a 0 in the denominator.

F

Determine the slant asymptote for the function f(x) = 3x²-4x+5/x-3.

y = 3x + 5

What is the solution of -8/2y-8 = 5/y+4 - 7y+8/y²-16?

y = 6

Which statement is true about the discontinuities of the function f(x)?

There are asymptotes at x = 7 and x = -4/3.

Sarah plans to sell homemade birdhouses at the local craft fair. She spent $90 on new tools and estimates that she spends $6 in materials for each birdhouse. Which rational function models the average cost per birdhouse, y, in dollars of building x birdhouses?

y = 90/x y = 90(x+6)/x _y = 90+0.6x/x y = 90/6x