Where is the function f(x)=3x^2-6x-45/x^2-5x discontinuous, and what types of discontinuities does it have?
Answers
Answer:
Below.
Step-by-step explanation:
f(x)=3x^2-6x-45/x^2-5x
= 3x^2-6x-45 / (x(x - 5))
The denominator is zero when x - 0 and x = 5.
So there will be a vertical asymptote at x = 0 and a hole at x = 5.
If we do the division we get f(x) = 3 remainder 9x - 45
so there will also be a horizontal asymptote where x = 3.
Answer:
Hole at (3, 4.8)
Asymptotes at x = 0 and y = 3
Step-by-step explanation:
[tex]f(x)=\dfrac{3x^2-6x-45}{x^2-5x}[/tex]
Factor the numerator:
[tex]\implies 3x^2-6x-45[/tex]
[tex]\implies 3(x^2-2x-15)[/tex]
[tex]\implies 3(x^2+3x-5x-15)[/tex]
[tex]\implies 3(x(x+3)-5(x+3))[/tex]
[tex]\implies 3(x-5)(x+3)[/tex]
Factor the denominator:
[tex]\implies x^2-5x[/tex]
[tex]\implies x(x-5)[/tex]
Factored form of function:
[tex]\implies f(x)=\dfrac{3(x-5)(x+3)}{x(x-5)}[/tex]
Discontinuity: a point at which the function is not continuous.
Holes
After factoring the rational function, if there is a common factor in both the numerator and denominator, there will be a hole at that point (x-value):
[tex]\implies (x-5)=0 \implies x=5[/tex]
Factor out the common factor from the function and input the found value of x into the new function to find the y-value of the hole:
[tex]\implies f(x)=\dfrac{3(x+3)}{x}[/tex]
[tex]\implies f(5)=\dfrac{3(5+3)}{5}=4.8[/tex]
Therefore, there is a hole at (5, 4.8)
Asymptotes
To find the vertical asymptotes, set the denominator of the new (factored) function to zero and solve:
⇒ Vertical Asymptote at x = 0
As the degrees of the leading terms of the numerator and denominator are equal (both x²), the horizontal asymptote is equal to the ratio of the leading coefficients.
⇒ Horizontal Asymptote at y = 3
Related Questions
A jar contains 100 gumballs. There are 40 blue gumballs, 3 green gumballs, 27 red gumballs, and 30 yellow gumballs. What is the probability of randomly selecting a red gumball?
Answers
Answer:
27%
Step-by-step explanation:
Red gumballs / total = 27/100 = 27%
26.
ZT.
Q
R
(9x - 17)
(4x + 28)
T
S
Answers
Simplifying
9x + -17 = 4x + 28
Reorder the terms:
-17 + 9x = 4x + 28
Reorder the terms:
-17 + 9x = 28 + 4x
Solving
-17 + 9x = 28 + 4x
Solving for variable 'x'.
Move all terms containing x to the left, all other terms to the right.
Add '-4x' to each side of the equation.
-17 + 9x + -4x = 28 + 4x + -4x
Combine like terms: 9x + -4x = 5x
-17 + 5x = 28 + 4x + -4x
Combine like terms: 4x + -4x = 0
-17 + 5x = 28 + 0
-17 + 5x = 28
Add '17' to each side of the equation.
-17 + 17 + 5x = 28 + 17
Combine like terms: -17 + 17 = 0
0 + 5x = 28 + 17
5x = 28 + 17
Combine like terms: 28 + 17 = 45
5x = 45
Divide each side by '5'.
x = 9
Simplifying
x = 9
Find the distance between the two given points:
(-5,-2);(-6, -6)
Answers
(-1, -4) or 1 to the left and 4 down
explanation:
to find the distance between two points on a coordinate plane, you have to subtract. (x_2 - x_1), (y_2 - y_1). -6 - -5 = -1 and -6 - -2 = -4. the second point is 1 to the left and 4 down from the first.
CORRECT ANSWER ONLY! I'LL GIVE BRAINLIEST!
Which graph represents the function f(x)= sqrt x − 2 - 2?
Answers
Answer:
the third fuction is the represents po
Step-by-step explanation:
and the next slide is the other one
eeeeeehelpmeeeeeeeeeee value of x and y
Answers
Step-by-step explanation:
As we can see x and 70° is alternate with each other so it can be name as alternative angle ....
Alternative angle are always equal....So .....
x = 70° ( being alternative angle )
Now....we can see that y and x is vertically opposite ....so it can be name as vertically opposite angle....
y = x ( being vertically opposite angle )
y = 70°
A sail shaped like a right triangle is shown. : Find the unknown side length. Enter the correct answer in the box. : Do these lengths form a Pythagorean triple? PLEASE HELP ME ASAP!!
Answers
Answer:
5 feet
Step-by-step explanation:
AB^2=AC^2+BC^2
13^2=AC^2+12^2
AC^2=25
AC=5
Pls help me with this i am giving 50pts out
Answers
20 points. brianliest if you get it right, and my appreciation
Answers
Answer:
x = 17
Step-by-step explanation:
75° and (4x + 7)° are alternate angles.
75 = 4x + 74x = 68x = 17Estimate the product of 67 x 23
by rounding to the nearest ten.
Answers
Answer:
1400
Step-by-step explanation:
67 rounds to 70 because 7 is bigger than 5 so it rounds up. 23 rounds down since 3 is smaller than 5.
So, 70x20=1400
A major fishing company does its fishing in a local lake. The first year of the company's operations it managed to catch 154,000 fish. Due to population decreases, the number of fish the company was able to catch decreased by 5% each year. How many total fish did the company catch over the first 10 years, to the nearest whole number?
Answers
Answer:
77,000 fish in 10 years
Step-by-step explanation:
calculator
Write each expression with a single rational exponent. 1/x^-1
Answers
[tex]~\dfrac 1{x^{-1}}\\\\=\dfrac{x^0}{x^{-1}}\\\\=x^{0-(-1)}\\\\=x^{0+1}\\\\=x^1\\\\=x[/tex]
Find the perimeter
................
Answers
Answer:
119 5/7
Explanation:
I’ve attached my worK
2. Answer each of the following based on the scenario below.
Scenario: You bought a car this year for $13,000. Unfortunately your car loses 25% of
-
its value every year.
a.
Does this scenario represent exponential growth or decay? Decay
b. What is the initial value of your savings account?$13,000
aishof wolse Hadapterit
c. What is the growth/decay factor? 25
d. Create a function in exponential form that models how much your car will be worth after x
year.
f(x) =
e. After 3 years how much will your car be worth?
f. After 8 years how much will your car be worth?
Answers
correct, is Decay and the initial amount is 13000, rate is 25%
[tex]\qquad \textit{Amount for Exponential Decay} \\\\ A=P(1 - r)^x\qquad \begin{cases} A=\textit{current amount}\\ P=\textit{initial amount}\dotfill &13000\\ r=rate\to 25\%\to \frac{25}{100}\dotfill &0.25\\ x=years \end{cases} \\\\\\ A=13000(1 - 0.25)^{x}\implies \boxed{A=13000(0.75)^x} \\\\\\ \stackrel{\textit{after 3 years}}{A=13000(0.75)^3}\implies A\approx 5484.38~\hfill \stackrel{\textit{after 8 years}}{A=13000(0.75)^8}\implies A\approx 1301.47[/tex]
which shows equation?
Answers
:)
Find the mean of the data set: 5, 19, 11, 3, 21, 9. (Round your answers to one decimal place, if necessary.)
Answers
Answer:
11.33
Step-by-step explanation:
Add all of the numbers together, then divide them by the number of numbers (6)
Plss help meee!!!!!!!!!!!!!
Answers
Answer:
Step-by-step explanation:
m<2 + 108 = 180
m<2 = 72
-2x^2 + 4x + 16 in vertex form
Answers
Answer:
y = - 2(x - 1)² + 18
Step-by-step explanation:
- 2x² + 4x + 16 ← factor out - 2 from the first 2 terms
= - 2(x² - 2x) + 16
using the method of completing the square
add/subtract ( half the coefficient of the x- term)² to x² - 2x
= - 2(x² + 2(- 1)x + 1 - 1) + 16
= - 2(x - 1)² + 2 + 16
= - 2(x - 1)² + 18 ← in vertex form
Cyclone filled 250 bottles of water to give people in the park. he handed out 40 bottles of water each hour he had 10 bottles left how many hours has cyclone been at the park
Answers
Which of these is a dependent variable and which of these is an independent variable.
I did Jumping Jacks
I spent 1 minute on the jumping jacks
I did 68 Jumping jacks in 1 minute.
BRAINLIEST, A LIKE, 5 STARS, 100 POINTS
Answers
Answer:
I did jumping jacks
Step-by-step explanation:
conditioning exercise performed from a standing position by jumping to a position with our legs spread and arms raised and then to the normal position.
Answer:
I hope nothing
hope. it helps
log64n=−31
step bye step on how to solve this
Answers
You get n= -17.163295
What is the area of the base of a cylinder with a volume of 2407 in.3 and a height of 16 inches?
Can someone PLS help me?
Answers
Answer:
15π in²
Step-by-step explanation:
Formula
V = πr²hSolving
240π = πr² × 16πr² (Area) = 15π in²What is the mean of 89, 73, 84, 91, 87, 77, 94
Answers
Answer:
85
Step-by-step explanation:
Mean = Sum of the terms / Number of terms
Mean = 89 + 73 + 84 + 91 + 87 + 77 + 94 / 7Mean = 595/7Mean = 85Calculating the relative frequencies from the data given in the table. Choose all that correctly describe an association between favorite subject and grade.
A higher percentage of 8th graders than 7th graders prefer History.
A higher percentage of 7th graders than 8th graders prefer English.
A higher percentage of 8th graders than 7th graders prefer Math/Science.
Answers
The statement second a higher percentage of 7th graders than 8th graders prefer English and statement third a higher percentage of 8th graders than 7th graders prefer Math/Science are correct.
What is a conditional relative frequency?It is defined as the frequency that can be evaluated by the two-way frequency table. Usually, we can obtain this frequency by dividing the frequency that is not in the total frequency cell to total frequency.
We can find conditional relative frequency as:
For first cell:
= 38/116
= 0.327
= 32.75% (refer attached picture)
Similarly, we can find for other cells.
A higher percentage of 7th graders than 8th graders prefer English and History, while a higher percentage of 8th graders than 7th graders prefer Math/Science.
Thus, the statement second a higher percentage of 7th graders than 8th graders prefer English and statement third a higher percentage of 8th graders than 7th graders prefer Math/Science are correct.
Learn more about the conditional relative frequency here:
brainly.com/question/8358304
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hey can anyone help me find the area pls ty
Answers
Answer:
42 . 9 in^2
Step-by-step explanation:
Area of trapezoid average of two bases X height
area = ( 3.3 + 9.9)/2 * 6.5 = 42.9 in^2
Find the horizontal and vertical shift of y=Sqrt x+2-3
Answers
Answer: Left 2, Down 3
Step-by-step explanation:
[tex]y=\sqrt{x+2}-3[/tex]
To find the horizontal shift, set the values under the root equal to 0
x + 2 = 0
x = -2
This means the functions is shifted left 2 units
To find the vertical shift, look at the -3
This means the graph is shifted down 3 units.
What is the value of x 2/3x+2=4
Answers
To solve for x, we can perform inverse operations to cancel values out.
For addition, we can do subtraction.For multiplication, we can do division.And vise versa.Eventually, we must isolate x to one side of the equal sign, or equation. That will give us its value.
Solving the Question[tex]\dfrac{2}{3}x+2=4[/tex]
⇒ First, isolate the x term by subtracting 2 from both sides:
[tex]\dfrac{2}{3}x=2[/tex]
⇒ Now, multiply both sides by 3 to get rid of the fraction:
[tex]2x=6[/tex]
⇒ Finally, divide both sides by 2 to isolate x:
[tex]x=3[/tex]
Answer[tex]x=3[/tex]
Answer:
x = 3
Step-by-step explanation:
[tex]\sf \dfrac{2}{3}x+2=4[/tex]
Subtract 2 from both sides.
[tex]\sf \dfrac{2}{3}x=4-2\\\\\sf \dfrac{2}{3}x=2[/tex]
Multiply both sides by 3.
[tex]\sf2x=2*3\\2x=6[/tex]
Divide both sides by 2.
[tex]\sf x=\dfrac{6}{2} \\\\\sf x=3[/tex]
Spencer bought 8 souvenirs during 2 days of vacation. How many days will Spencer have to spend on vacation before he will have bought a total of 12 souvenirs? Assume the relationship is directly proportional
Answers
Answer:
3 days
Step-by-step explanation:
hope u understand
newton watches a movie with his friends. they watch 50% of the movie and then takes a break. they then watch the remaining 65 minutes = how long was the movie
Answers
Answer:
130 minutes
Step-by-step explanation:
They watched 50% of movie, means 50% left unwatched
Remaining length = 65 minutes = 50% of movie
Movie length = 65 *2 = 130 minutes
If they watched 50% which is half you take 65 and multiply it by 2= 130
HELP
1. (08.01) Two lines, A and B, are represented by the following equations:
Line A: y = -x + 4
Line B: y = -x + 4
Which statement is true about the solution to the set of equations? (4 points)
It is (1, 2).
There are infinitely many solutions.
It is (1, 5).
There is no solution.
Answers
Answer:
there is no solution since the equations are the same hence on trying to solve all variables will cancel out
please help what is the effect on the graph of the parent function f(x) = x when f(x) is replaced with 4f(x)
Answers
A function assigns the values. When in the function f(x)=x, f(x) is replaced by 4f(x), the slope of the function reduces by a factor of 4.
What is a Function?A function assigns the value of each element of one set to the other specific element of another set.
Given the function f(x) is equal to x. If f(x) in the given function is replaced with 4f(x), then the result will be,
[tex]4f(x) = x\\\\f(x)=\dfrac{x}4 \\\\ f(x) = \dfrac14 \times x[/tex]
Hence, when in the function f(x)=x, f(x) is replaced by 4f(x), the slope of the function reduces by a factor of 4.
Learn more about Function:
https://brainly.com/question/5245372
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