Where do you find your domain restrictions in a rational function?
Answers
Answer:
denominator zeros are excluded
Step-by-step explanation:
The domain is the set of values of the independent variable where the function is defined. A rational function is "undefined" where the denominator is zero.
The exclusions of interest are generally those values of the variable that make any denominator be zero.
__
Example:
f(x) = (2/(x-3)) / (4/(x -6))
has denominators of x-3 and x-6. These are zero for x=3 and x=6, so those two values are excluded from the domain of this function. We observe that the function can be simplified to ...
f(x) = (x -6)/(2(x -3))
but x=6 remains excluded from the domain because of its effect in the original function definition.
_____
Additional comment
If the rational function includes functions that have domain restrictions (such as square root, for example), then those restrictions apply as well.
Usually, but not always, the rational functions where you're asked about domain are the ratios of polynomials. So, you need to factor or otherwise find the zeros of any denominator polynomials in such functions.
Related Questions
This stem and leaf diagram shows the number of students who go to various after school clubs. What is the smallest number of students who go to one of these clubs?
1| 6 8 9
2| 1
3| 5 9
4| 0 2 4 5
(Key: 2| 1 represents 21 students)
Answers
The smallest number of students who go to one of these clubs is 21.
A stem-and-leaf diagram is a quantitative assessment of a collection of data in which the data values are divided into a "stem" (the first digit of the number) and a "leaf" (the remainder of the number) (the last digit of the number).
Based on the given stem and leaf diagram, the smallest number of students who go to one of these clubs is represented by the leaf value "1" in the stem "2."
Therefore, the smallest number of students who go to one of these clubs is 21.
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Choose the inequality that represents the following graph
Answers
Answer:
X ≥ 3 (D)
Step-by-step explanation:
I got it right on Khan Academy :)
If BH = 3, BF = 10, GH = 2, find HD
Answers
Answer:
HD = 10.5
Step-by-step explanation:
Given BH = 3, GH = 2, BF = 10
Step 1: To find HF:
HF = BF – BH
HF = 10 – 3
HF = 7
Step 2: To find HD:
We know that if two chords intersects inside a circle, the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.
⇒ GH × HD = BH × HF
⇒ 2 × HD = 3 × 7
⇒ HD = 10.5
Hence, the value of HD = 10.5.
I hope this helps a little bit.
Answer:
HD= 10.5
Step-by-step explanation:
it's 10.5
A car is traveling at a rate of 80 miles per hour. What is the car's rate in kilometers per hour? How many kilometers will the car travel in 5 hours? In your computations, assume that 1 mile is equal to 1.6 kilometers. Do not round your answers. Rate: Distance traveled in 5 hours:
Answers
The car will travel 640 kilometers in 5 hours.
We have,
To convert miles per hour to kilometers per hour, we need to multiply by the conversion factor of 1.6:
= 80 miles/hour x 1.6 kilometers/mile
= 128 kilometers/hour
So the car's rate is 128 kilometers per hour.
Now,
To find the distance the car will travel in 5 hours, we need to multiply the rate by the time:
= 128 kilometers/hour x 5 hours
= 640 kilometers
Thus,
The car will travel 640 kilometers in 5 hours.
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-3/5(1+5n) =102
Solve for n
Answers
Answer:
Step-by-step explanation:
Answer:
Step-by-step explanation:
2. A car costs $10,350 and decreases in value by 1% per month. How much will the car be worth
after 3 years?
Answers
Answer:
The car will be worth 6,624$ in 3-years
Step-by-step explanation:
Firstly we need to move the decimal two to the left, so the decimal form of 1% is 0.01.
Secondly, we make the equation by plugging in a multiplication sign and the cost of the car. Example - (0.01 x 10,350).
Now, that we have everything in an equation we can multiply 0.01 by 10,350, which will give us what 0.01 percent of 10,350 is, (103.5).
Lastly, we will subtract 103.5 from the cost of the car (10,350) to get the value of the car after the first month. The cost of the car will be $10,246.5. To find the value after the next two years we will repeat these steps...
The ratio of 1.2 to 29 is the same as the ratio of 4.8 to
Answers
Answer:
116
Step-by-step explanation:
Given the ratio 1.2 : 29, you want x in the equivalent ratio 4.8 : x.
SetupEquivalent ratios can be written as a proportion:
1.2/29 = 4.8/x
SolutionMultiplying by 29x, we have ...
1.2x = 29(4.8)
Dividing by 1.2 gives ...
x = 29(4.8/1.2) = 29(4) = 116
The ratio is the same as 4.8 to 116.
__
Additional comment
A proportion like this can be written 4 ways. The solution takes only one step if the unknown is in the numerator.
\(\dfrac{1.2}{29}=\dfrac{4.8}{x},\quad\dfrac{29}{1.2}=\dfrac{x}{4.8},\quad\dfrac{1.2}{4.8}=\dfrac{29}{x},\quad\dfrac{4.8}{1.2}=\dfrac{x}{29}\)
The equation represents the function f, and the graph represents the function g.f(x) = 6x - 5Determine the relationship between the rates of change of f and g
Answers
Solution:
Given that;
The equation represents the function f, and the graph represents the function g.
Where
\(f(x)=6x-5\)To find the slope of g(x), we will pick points on the graph
\(\begin{gathered} (x_1,y_1)=(0,2) \\ (x_2,y_2)=(-1,-1) \end{gathered}\)The formula to find slope, m, is
\(m=\frac{y_2-y_1}{x_2-x_1}\)Substitute the values of the variables into the formula above
\(\begin{gathered} m=\frac{-1-2}{-1-0}=\frac{-3}{-1}=3 \\ m=3 \end{gathered}\)The slope of g(x) is 3
The slope of f(x) is 6,
Where,
Rate of change is the same as the slope of the function/graph
The relationship between the rates of change of f and g will be
\(\begin{gathered} \frac{Rate\text{ of change of f\lparen x\rparen}}{Rate\text{ of change of g\lparen x\rparen}}=\frac{6}{3}=\frac{2}{1} \\ Rate\text{ of change of f\lparen x\rparen}:Rate\text{ of change of g\lparen x\rparen}=2:1 \end{gathered}\)Hence, the rate of change of f is twice the rate of change of g (option D)
Find the weight of a 176-cm male to the nearest kg whose BSA = 2.3.
Answers
The weight of the male that has a body surface area of 2.3 would be = 108kg
What is Body Surface Area (BSA)?The body surface area (BSA) is defined as the parameter that can be used to estimate the surface area of a person's body based on body weight and height.
The formula for BSA = √w×h/3600
The weight of the male = X
The height of the male = 176cm
The BSA = 2.3
From the formula, make w the subject of formula;
w = BSA²× 3600/h
w = 2.3²× 3600/176
w= 5.29×3600 /176
w = 19044/176
w = 108kg
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If mC=6x+4 and mA=2x + 12, solve for x
Answers
Answer:
x = 2
Step-by-step explanation:
It is the picture of parallelogram . And in a parallelogram , vertex angles which are opposite to each other are equal. So, angleA = angleC and angleB = angleD.
As angleA = angleC ,
\(6x + 4 = 2x + 12\)
\( = > 6x - 2x = 12 - 4\)
\( = > 4x = 8\)
\( = > x = \frac{8}{4} = 2\)
A negative number on the x-axis (-a, b) would move in what direction?
A positive number on the x-axis (+a, b) would move in what direction?
A negative number on the y-axis (a, -b) would move in what direction?
A positive number on the y-axis (a, +b) would move in what direction?
Please answer for points and brainliest!
Answers
a) A negative number on the x-axis (-a, b) would move in the left direction by one unit.
To find out why, check point (0, b) on the y-axis and point (-a, b) on the x-axis.
The distance between these two points is a unit, meaning that the point (-a, b) is one unit to the left of the point (0, b).
b) A positive number on the x-axis (+a, b) would move in the right direction by a unit.
Again, let's look at the point (0, b) on the y-axis and the point (+a, b) on the x-axis.
The distance between the two points is also one unit, which means the point (+a, b) is one unit to the right of the point (0, b).
c) A negative number on the y-axis (a, -b) would move in the downward direction by b units.
Assume that point (a, 0) is on the x-axis and point (a, -b) is on the y-axis. The distance between them is b units, which means that the point (a, -b) is b units below the point (a, 0).
d) A positive number on the y-axis (a, +b) would move in the upward direction by b units.
Also, check the point (a, 0) on the x-axis and point (a, +b) on the y-axis. The distance between these two points is b units, which means that the point (a, +b) is b units above the point (a, 0).
What is a number?A number is a mathematical term used to show the quantity or value of a thing. It can be depicted using numerals, symbols, or words.
Examples include 30, hundred, -8, 6x, "5", 0.67, etc.
Numbers can be Positive - numbers greater than zero, or negative - numbers less than zero.
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I need to find what the square root is to 19,478,730. my teacher literally put that. help
Answers
Answer: 4413.5
Step-by-step explanation:
4413.47142 if you need to be exact
A cube has edge length 3 in. What is its volume?
Answers
The volume of a cube is given by the formula V = s^3, where s is the length of one of its sides. So, if the edge length of a cube is 3 inches, its volume would be:
V = 3^3 = 27 cubic inches
Question 2(Multiple Choice Worth 2 points)
(Slope-Intercept Form MC)
The table shown represents a linear relationship.
x 0 1 3 4
y −8 −6 −2 0
Based on the table, what is the equation of the linear relationship in slope-intercept form?
y = 2x − 8
y = 2x + 8
y = −2x + 4
y = −2x − 4
Answers
The equation of the linear relationship in slope-intercept form is y = 2x - 8. Option A is the correct answer.
To determine the equation of the linear relationship in slope-intercept form based on the table, we need to find the slope and y-intercept.
By observing the table, we can calculate the slope by selecting any two points. Let's choose the points (0, -8) and (4, 0).
Slope (m) = (change in y) / (change in x)
= (0 - (-8)) / (4 - 0)
= 8 / 4
= 2
Now that we have the slope, we can find the y-intercept by substituting the values of one point and the slope into the equation y = mx + b and solving for b.
Using the point (0, -8):
-8 = 2(0) + b
b = -8
Therefore, the equation of the linear relationship in slope-intercept form is: y = 2x - 8. Option A is the correct answer.
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Select the postulate that specifies the minimum number of points in space.
Postulate 1: A line contains at least two points.
Postulate 1a: A plane contains at least three points not all on one line.
Postulate 1b: Space contains at least four points not all on one plane.
Postulate 2: Through any two different points, exactly one line exists.
•
Postulate 3: Through any three points that are not one line, exactly one plane exists.
Postulate 4: If two points a lie in a plane, the line containing them lies in that plane.
Postulate 5: If two planes intersect, then their intersection is a line.
Answers
The postulate specifies the minimum number of points in space is, Postulate 1b: Space contains at least four points not all on one plane.
A postulate is an assumption or a claim that is made without providing any supporting evidence. The fundamental claims that are used to support other claims known as theorems are known as postulates. Once a theorem is established, it can be applied to the establishment of other theorems.
By examining each postulate, we may effectively respond to this query.
Postulate 1a: establishes that if G and H are distinct points on plane R, then R contains a third point rather than GH.
Postulate (1b): establishes the bare minimum within a space.
Postulate 1: This describes a line segment with two points, say A and B.
Postulate 2: It asserts that two points determine a line.
Postulate 3: This one explains why, for example, if one leg is shorter than the other three, a table with four legs would wobble, but a table with three legs would not wobble.
Postulate 4: When points A and B are in Q, it is confirmed that AB is in plane Q.
Postulate 5: It concerns two planes.
According to the data presented above, the appropriate response to our query is:
Postulate 1b: Space contains at least four points not all on one plane.
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The revenue function R in terms of the number of units sold, a, is given as R= 380x - 0.1x^2where R is the total revenue in dollars. Find the number of units sold a that produces a maximum revenue?Your answer is x= What is the maximum revenue? __
Answers
The maximum revenue corresponds to the y-coordinate of the vertex of the graph of the revenue function. And the number of units sold that produces a maximum revenue corresponds to the x-coordinate of the vertex of the graph of the revenue function.
Then, we can find the vertex of the revenue function. For this, we can rewrite the function in its vertex form by completing the square.
\(\begin{gathered} f(x)=ax^{^2}+bx+c\Rightarrow\text{ General form} \\ f(x)=a(x-h)^2+k\Rightarrow\text{ Vertex form} \\ \text{ Where }(h,k)\text{ is the vertex} \end{gathered}\)Then, we have:
Step 1: Reorder the terms.
\(\begin{gathered} R=380x-0.1x^2 \\ R=-0.1x^2+380x \end{gathered}\)Step 2: We use the general form to find the values of a,b and c.
\(\begin{gathered} a=-0.1 \\ b=380 \\ c=0 \end{gathered}\)Step 3: We find the value of h using the below formula.
\(h=\frac{-b}{2a}\)\(\begin{gathered} h=\frac{-380}{2(-0.1)} \\ h=\frac{-380}{-0.2} \\ h=1900 \end{gathered}\)Step 4: We find the value of k using the below formula.
\(k=c-\frac{b^2}{4a}\)\(\begin{gathered} k=0-\frac{380^2}{4(-0.1)} \\ k=\frac{-144400}{-0.4} \\ k=361000 \end{gathered}\)Step 5: We substitute the values of a, h and k into the vertex form.
\(\begin{gathered} \begin{equation*} f(x)=a(x-h)^2+k \end{equation*} \\ R=-0.1(x-1900)^2+361000 \end{gathered}\)Thus, the vertex of the revenue function is the ordered pair (1900,361000).
AnswerThe number of units sold that produces a maximum revenue is 1900, and the maximum revenue is $361000.
The speed of the wind is changing according to the function V(t)=-1/2+30, where t=time in minutes after 6:00 p.m and V(t) = wind speed in miles per hour (mph). Which statement best describes the meaning of the equation V(20)-V(10)=-5? .
Answers
Answer: D
Step-by-step explanation: In a period of 5 minutes. the wind speed decreases from 20mph to 10mph
please help.. no links:)
Answers
Answer:
Step-by-step explanation:
68. A table top and the floor are a model of parallel planes.
a. True
b. False
Answers
Answer:
true
Step-by-step explanation:
True the table top can go on forever without touching the floor.
Verify that each equation is an identity.
Show Work plzz!!
Answers
Answer:
Step-by-step explanation:
sin² x + cos² x = 1
\(\frac{1}{cosx}\) = sec x
( \(\frac{1}{cosx}\) )² = sec² x
PLZ ANSWER IF U KNOW THE ANSWER
Maria School Bus field 4 5/8 buses to take people on a trip to the county pool. each bus holds 72. people how many people rode the buses on the trip?
Answers
Aaron bought 2 pounds of cheese for $6. Assuming the situation is proportional, determine how much it will cost for Aaron to buy 15 pounds of cheese.
Answers
help with the question please
Answers
For the question of total area of the cuboid is 200cm^.
I understand where we divide 150 by 4.
But why do I need to multiply by 5, when there are 6 faces.
Answers
You need to multiply by 5 instead of 6 because each pair of opposite faces on a cuboid has the same area, so by considering one face from each pair, you ensure that you don't count any face twice.
When calculating the total surface area of a cuboid, you need to understand the concept of face pairs.
A cuboid has six faces, but each face has a pair that is identical in size and shape.
Let's break down the reasoning behind multiplying by 5 instead of 6 in the given scenario.
To find the surface area of a cuboid, you can add up the areas of all its faces.
However, each pair of opposite faces has the same area, so you avoid double-counting by only considering one face from each pair. In this case, you have five pairs of faces:
(1) top and bottom, (2) front and back, (3) left and right, (4) left and back, and (5) right and front.
By multiplying the average area of a pair of faces by 5, you account for all the distinct face pairs.
Essentially, you are considering one face from each pair and then summing their areas.
Since all the pairs have the same area, multiplying the average area by 5 gives you the total surface area.
When dividing 150 by 4 (to find the average area of a pair of faces), you are essentially finding the area of a single face.
Then, by multiplying this average area by 5, you ensure that you account for all five pairs of faces, providing the total surface area of the cuboid.
Thus, multiplying by 5 is necessary to correctly calculate the total surface area of the cuboid by accounting for the face pairs while avoiding double-counting.
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The theater sells two types of tickets: adult tickets for $6 and child tickets for $4.
Last night, the theater sold a total of 364 tickets for a total of $1930. How many adult tickets did the theater sell last night?
Answers
Answer:
237
Step-by-step explanation:
This is a system of equations.
The theater sold 364 adult and child tickets, so a + c = 364
They made a total of $1930. Each adult ticket was $6 & child tickets were $4. The second equation is 6a + 4c = 1930.
Let's line them up
a + c = 364
6a + 4c = 1930
Since we need to solve for the number of adult tickets, we want to get rid of the c variable. I'm going to multiply the entire first equation by -4 to do this. The second equation stays the same. Now, I have:
-4a - 4c = -1456
6a + 4c = 1930 Add them together
----------------------
2a = 474 Divide by 2 to solve for a
a = 237
There were 237 adult tickets sold
The height of right circular cylinder P
is twice the height of right circular
cylinder Q. The radii of the cylinders are
of equal length
What number times the volume of
cylinder Q is equal to the volume of
cylinder P?
A. 2
B. 4
C. 6
D. 8
Answers
Answer:
A. 2
Step-by-step explanation:
The computation is shown below:
As we know that
The Volume of a right circular cylinder is
\(V = \pi r^h\\\\\)
Here r is the radius
And h is the height
Now it is mentioned that the height of the right circular cylinder P is double to the height of the right circular cylinder Q
Now let us assume h be the height of cylinder p
And, H be the height of cylinder Q
So the equation is
h = 2H ........(1)
Also
The radius of both the cylinders would be the similar length
So
we assume the r be the radius of both cylinders
Now
The Volume of cylinder Q = \(V_Q = \pi r^2H\)
And for P it is \(V_p = \pi r^2 h\)
Now substitute equation 1
\(V_p = \pi r^2(2H)\\\\V_p= 2 \pi r^2hH\\\\V_p = 2(\pi r^2H)\\\\V_p = 2(V_Q)\)
Hence, the correct option is A.
uppose a large telephone manufacturer has a problem with excessive customer complaints and consequent returns of the phones for repair or replacement. The manufacturer wants to estimate the magnitude of the problem in order to design a quality control program. How many telephones should be sampled and checked in order to estimate the proportion defective to within 4 percentage points with 83% confidence
Answers
Answer:
The answer is "294.2075".
Step-by-step explanation:
Given:
\(\hat{p} = 0.5 \\\\1 - \hat{p} = 1 - 0.5 = 0.5\\\\E = 4\% = 0.04\\\\\)
At \(83\%\) confidence level the z is ,
\(\alpha = 1 - 83\% = 1 - 0.83 = 0.17\\\\\frac{\alpha}{2} = \frac{0.17}{2} = 0.085\\\\Z_{\frac{\alpha}{2}} = Z_{0.085} = 1.3722\\\\n = (\frac{Z_{\frac{\alpha}{2}}}{E})^2 \times \hat{p} \times (1 - \hat{p})\\\\\)
\(= (\frac{1.3722}{0.04})^2 \times 0.5 \times 0.5\\\\= (34.305)^2 \times 0.5 \times 0.5\\\\= 1,176.83 \times 0.5 \times 0.5\\\\= 1,176.83 \times 0.25\\\\=294.2075\)
The Corner Cafe offers a 10% off senior citizen discount. If Mr. and Mrs. Schmidt order $36.00 worth of food and are senior citizens, how much can they expect to save?
Answers
Answer:
$3.60
Step-by-step explanation:
$36.00 × 10%= $3.60
Match each linear system with one of the phase plane direction fields.
(The blue lines are the arrow shafts, and the black dots are the arrow tips.)
I need help. I don't know how to use the eigenvalues and take that information and put it on a graph. Please explain thoroughly! Thanks.
Answers
The correct match for each linear system with one of the phase plane direction fields will be:
1:B
2:C
3:D
4:A
What is plane?
In geometry, a plane is a surface made up of all the straight lines connecting any two locations on it. It is, in other words, a level or flat surface. A plane is uniquely defined in a Euclidean space of any number of dimensions by one of the following: three non-collinear points are used. Other names for it include a two-dimensional surface. A plane has an unlimited width, an endless length, zero thickness, and no curvature. The flat surfaces of a cube or cuboid, as well as the flat surface of paper, are all true instances of geometric planes, despite the fact that it is difficult to envision a plane in real life.
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Triangles P and Q are similar.
Find the value of XZ.
C
12m
X
P
5m
B
Q
The diagram is not drawn to scale.
xe m
N
22.5m
Answers
The missing side length XZ in the figure is 54 cm
How to find the side length XZ in the triangleFrom the question, we have the following parameters that can be used in our computation:
The similar triangles
To calculate the side length, we make use of the following equation
XZ : 22.5 = 12 : 5
Where the missing length is represented with XZ
Express as a fraction
So, we have
XZ/22.5 = 12/5
Next, we have
XZ = 22.5 * 12/5
Evaluate
XZ = 54
Hence, the missing side length XZ is 54 cm
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