1. Find two functions f(x) and g(x) such that f[g(x)] = x but g[f(x)] does not equal x.2. State the vertical, horizontal asymptotes and zeros of the rational function, f(x) = {x^2+3x+2}/{x^2+5x+4}.Why is there no zero at x = –1?3. Give an example and explain why a polynomial can have fewer x-intercepts than its number of roots.
Answers
Answer:
1. [tex]f(x)=x^{2}[/tex] and [tex]g(x)=-\sqrt{x}[/tex]
2. Vertical Asymptote: x=-4
Horizontal Asymptote: y=1
Zeros: (-2,0)
Because the graph will have a hole at x=-1. This is, the graph is not defined for that particular x-value.
3. [tex]f(x)=x^{2}+4[/tex]
It has two complex roots, so there is no place in the graph where it crosses the x-axis.
Step-by-step explanation:
1. If we find the composite function f[g(x)] with the functions:
[tex]f(x)=x^{2}[/tex] and g(x)=-x, we will get the following:
[tex]f[g(x)]=(-\sqrt{x})^{2}[/tex]
which simplifies to
f[g(x)]=x
on the other hand, if we find g[f(x)] we will get the following:
[tex]g[f(x)]=-\sqrt{x^2}[/tex]
which simplifies to:
g[f(x)]=-x
2.
Before finding the asymptotes, it's a good idea to factor both the numerator and denominator of the function, so we get:
[tex]f(x)=\frac{x^{2}+3x+2}{x^{2}+5x+4}[/tex]
[tex]f(x)=\frac{(x+2)(x+1)}{(x+4)(x+1)}[/tex]
we can now see that this function can be simplified. It is important to simplify this function so we don't confuse vertical asymptotes with holes in the graph, so when simplifying we get the following:
[tex]f(x)=\frac{x+2}{x+4}[/tex]
now we can set the denominator equal to zero.
x+4=0
and solve for x:
x=-4 this is our vertical asymptote.
For the horizontal asymptote, there is a rule that tells us that if the polynomials on the numerator and denominator have the same degree (the same greatest power), then the horizontal asymptote is found by dividing the lead coefficient of the polynomial on the numerator by the lead coefficient on the denominator. In this case the lead coefficients are 1 and 1 so the horizontal asymptote is:
[tex]y=\frac{1}{1}[/tex]
so the horizontal asymptote is at:
y=1.
The zeros of the function are located at the x-values that will turn the numerator of the simplified fraction equal to zero, so we get:
x+2=0
x=-2
so the only zero will be located at (-2,0)
x=-1 is not a zero because the original function is not defined for x=-1, there is a hole in the graph at this x-value.
Take a look at the graph of the function (See attached picture)
3.
This will happen when there are no real zeros. Take for example the function:
[tex]f(x)=x^{2}+4[/tex]
In this case this problem has 2 non-real zeros:
[tex]x^{2}+4=0[/tex]
[tex]x^{2}=-4[/tex]
[tex]x=\sqrt{-4}[/tex]
so:
x=2i or x=-2i
since they are complex numbers they cannot be graphed on a coordinate axis, so the graph will not cross the x-axis. See attached picture.
![1. Find Two Functions F(x) And G(x) Such That F[g(x)] = X But G[f(x)] Does Not Equal X.2. State The Vertical,](https://brainimage.s3.us-east-005.backblazeb2.com/answers/attachments/VMyjpYAMpYmUlTab2TH9KheiKXLjrxGW.png)
![1. Find Two Functions F(x) And G(x) Such That F[g(x)] = X But G[f(x)] Does Not Equal X.2. State The Vertical,](https://brainimage.s3.us-east-005.backblazeb2.com/answers/attachments/IIxvD1SnEnt5Og55tUsak5zZjljpO4H3.png)
Related Questions
A college library has five copies of a certain text on reserve. Three copies (1, 2, and 3) are first printings, and the remaining two (4, and 5) are second printings. A student examines these books in random order, stopping only when a second printing has been selected. One possible outcome is 4, and another is 214. (a) List the outcomes in S. (b) Let A denote the event that exactly one book must be examined. What outcomes are in A
Answers
Answer:
S = { (4), (5), (1,4), (1,5), (2,4), (2,5), (3,4), (3,5), (1,2,3,4), (1,2,3,5), (2,1,3,4) (2,1,3,5), (3,1,2,4), (3,1,2,5) }
A ={(4), (5)}
Step-by-step explanation:
Given that:
Among the three copies, (1,2,3) are the first printings, and (4,5) are the second printings.
A student who examines these books in random order stops when a second printing has been selected.
Thus, we can compute the sample space associated with these experiments as:
S = { (4), (5), (1,4), (1,5), (2,4), (2,5), (3,4), (3,5), (1,2,3,4), (1,2,3,5), (2,1,3,4) (2,1,3,5), (3,1,2,4), (3,1,2,5) }
Suppose A represents the event that we must examine exactly one book.
Then the outcomes of A are:
A ={(4), (5)}
Please help! If x = 6 and b = 4, what is x+b
Answers
Hey there!
ANSWER: [tex]10[/tex]EXPLANATION:If you want to know how to solve, you will need to find the value of x and y. x is equal to 6 and b is equal to 4.
[tex]x(6)+b(4)=?[/tex]
So we basically just add 6 and 4.
[tex]6+4=10\\x+b=10(Answer)[/tex]
Hope this helps!
[tex]-TetsedHyperr[/tex]
The formula for calculating the density of an object is D=-, where m is mass and v is volume.
m
Rewrite the formula in terms of m. Then rewrite the formula in terms of v.
Answers
Answer:
M= D x V, V= M x D
Step-by-step explanation:
You just do the opposite of division since the original is:
Density= Mass/ Volume
**LOOK AT PHOTO** Whats the answer and what are the steps?
Answers
Answer:
[tex]a=2K-b[/tex]
Step-by-step explanation:
[tex]K=\frac{a+b}{2}[/tex]
Switch sides:
[tex]\frac{a+b}{2}=K[/tex]
Multiply both sides by 2:
[tex]\frac{2\left(a+b\right)}{2}=2K[/tex]
[tex]a+b=2K[/tex]
Subtract b from both sides:
[tex]a+b-b=2K-b[/tex]
[tex]a=2K-b[/tex]
Solve y over negative 6 + 5 = 9. (1 point)
Answers
This should be your equation - (y/-6) + 5 = 9
Answer:
y/-6 = 4 <y = -24 <
Answer:
[tex]y=-24\\[/tex]
Step-by-step explanation:
Write the equation
[tex]\frac{y}{-6} +5=9[/tex]
Subtract 5 from both sides
[tex]\frac{y}{-6} =4[/tex]
Multiply both sides by -6
[tex]y=-24\\[/tex]
Hope this helps! : )
Rewrite the given inequality as two linear inequalities. ∣8x+9∣<4
Answers
Answer:
[tex]8x+9<4[/tex] and [tex]8x+9>-4[/tex].
Step-by-step explanation:
The given inequality is
[tex]|8x+9|<4[/tex]
We need to rewrite the given inequality as two linear inequalities.
We know that |8x+9|>0 but (8x+9) can be positive or negative.
For [tex]|8x+9|<4[/tex], the value of (8x+9) must be lies between -4 and 4.
i.e., [tex]-4<8x+9<4[/tex]
If 8x+9 positive, then
[tex]8x+9<4[/tex] ...(1)
If 8x+9 negative, then
[tex]8x+9>-4[/tex] ...(2)
Therefore, the two inequalities are [tex]8x+9<4[/tex] and [tex]8x+9>-4[/tex].
Calculate the distance between (6 − 5i) and ( 1 + 3i).
A: /113
B: /29
C: /53
D: /89
Answers
Answer: D
Step-by-step explanation:
i just had this question on a quiz and it said the correct answer was /89
can someone please help me with this?!
Answers
Answer:
26) Let the first number be x.
Sum of three consecutive even numbers:
[tex]x+(x+2)+(x+4)=-84[/tex]
[tex]x+x+2+x+4=-84\\[/tex]
[tex]3x+6=-84[/tex]
Subtract 6 from both sides:
[tex]3x+6-6=-84-6[/tex]
[tex]3x=-90[/tex]
Divide both sides by 3:
[tex]\frac{3x}{3}=\frac{-90}{3}[/tex]
[tex]x=-30[/tex]
To find the other two numbers, add 2 and 4 respectively:
[tex](-30+2=-28)(-30+4=-26)[/tex]
27) Let the first number be x.
Sum of three consecutive odd integers:
[tex]x+(x+2)+(x+4)=141[/tex]
[tex]3x+6=141[/tex]
Subtract 6 from both sides:
[tex]3x+6-6=141-6\\3x=135[/tex]
Divide both sides by 3:
[tex]\frac{3x}{3}=\frac{135}{3}\\x=45[/tex]
To find the other two numbers, add 2 and 4 respectively:
[tex](45+2=47)(45+4=49)[/tex]
28) Let the first number be x.
Sum of four consecutive integers:
[tex]x+(x+1)+(x+2)+(x+3)=54[/tex]
[tex]4x+6=54[/tex]
Subtract 6 from both sides:
[tex]4x+6-6=54-6\\4x=48[/tex]
Divide both sides by 4:
[tex]\frac{4x}{4}=\frac{48}{4}\\x=12[/tex]
To find the other three numbers, add 1, 2 and 3 respectively:
[tex](12+1=13)(12+2=14)(12+3=15)[/tex]
29) Let the first number be x.
Sum of four consecutive integers:
[tex]x+(x+1)+(x+2)+(x+3)=-142[/tex]
[tex]4x+6=-142\\[/tex]
Subtract 6 from both sides:
[tex]4x+6-6=-142-6[/tex]
[tex]4x=-148[/tex]
Divide both sides by 4:
[tex]\frac{4x}{4}=\frac{-148}{4}\\x=-37[/tex]
To find the other three numbers, add 1, 2 and 3 respectively:
[tex](-37+1=-36)(-37+2=-35)(-37+3=-34)[/tex]
Which of the following statements best describes a theorem?
A. A theorem cannot be proven.
B. A theorem can be false.
C. A theorem is always true.
D. A theorem is never true.
Answers
Answer:
a
Step-by-step explanation:
A theorem cannot be proven.
2,802,136 expanded complete the expanded form
Answers
Answer:
200000+800000+0+2000+100+30+6
Step-by-step explanation:
2,000,000
800,000
0
2,000
100
30
+ 6
_________
A soccer ball with a diameter of 8.6 inches is shipped in a box that is a square prism and has a side length of 9.5 inches. How much volume is available to be filled with packing material if the shipping company wants the box completely full? Round your answer to the nearest tenth.
524.5 in^3
774.2 in^3
5,115.8 in^3
1805.6 in^3
1190.2 in^3
Answers
Answer:
A 524.5 in^3
Step-by-step explanation:
A study was done by an online retail store to determine the rate at which users used its website. A graph of the data that was collected is shown: A line graph with Number of Months on the x axis and Number of Users, in thousands, on the y axis. The x axis has a scale from 0 to 36 with an increment of 4. The y axis has a scale of 0 to 60 with increments of 6. A straight line connecting 0, 0 and approximately 36, 54 is drawn. What can be interpreted from the range of this graph? v
Answers
Answer:
A graph of the data that was collected is shown: A line graph with Number of Months on the x axis and Number of Users, in thousands, on the y axis. The x axis has a scale from 0 to 36 with an increment of 4. The y axis has a scale of 0 to 60 with increments of 6. A straight line connecting 0, 0 and approximately 36, 54 is drawn. What can be interpreted from the range of this graph? v .
Answer:
c
Step-by-step explanation:
Write the word form of the following decimals. 432.98
Answers
Answer: Four hundred and thirty two point nine - eight.
Step-by-step explanation:
Hey there! I'm happy to help!
With the numbers before the decimal, you just say it normally. After the decimal, you just name the digits.
So, you would call this number four hundred thirty-two point nine eight.
Have a wonderful day! :D
Eight avocados cost $4.
How much do 16 avocados cost?
How much do 20 avocados cost?
How much do 9 avocados cost?
Answers
20 avocados cost $10
9 avocados cost $4.5
16 avocados costs = $8, 20 avocados costs = $10 and 9 avocados costs = $4.5
What is the Cost?The amount of money that a company spends on the creation or production of goods or services.
Given that, Eight avocados cost $4.
Therefore, each of them costs = 4/8 = 1/2
So, 16 avocados = 16*1/2 =$8
20 avocados = 20*1/2 =$10
9 avocados = 9*1/2 =$4.5
Hence, 16 avocados costs = $8, 20 avocados costs = $10 and 9 avocados costs = $4.5
For more references on Cost, click;
https://brainly.com/question/15135554
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Find the derivative of the function. y = sin−1(6x + 1)
Answers
If you don't know the derivative of the inverse of sine, you can use implicit differentiation. Apply sine to both sides:
[tex]y=\sin^{-1}(6x+1)\implies\sin y=6x+1[/tex]
(true for y between -π/2 and π/2)
Now take the derivative of both sides and solve for it:
[tex]\cos y\dfrac{\mathrm dy}{\mathrm dx}=6[/tex]
[tex]\dfrac{\mathrm dy}{\mathrm dx}=6\sec y[/tex]
[tex]\dfrac{\mathrm dy}{\mathrm dx}=\dfrac6{\cos\left(\sin^{-1}(6x+1)\right)}[/tex]
[tex]\dfrac{\mathrm dy}{\mathrm dx}=\dfrac6{\sqrt{1-(6x+1)^2}}[/tex]
HELP!!!
on thursday, mabel handled 90 transactions. anthony handled 10% more transactions than mabel. cal handled 2/3 of the transactions that anthony handled, and Jade handled 16 more transactions than cal. how many transactions did jade handle?
Answers
Answer:
82
Step-by-step explanation:
Mabel = 90Anthony = 90 + 10% = 90 +90*0.1 = 99Cal = 2/3*99 = 2*33 = 66Jade = 16 + 66 = 82Eli is making smoothies. Each smoothie uses 1/2 cup of yogurt. How much yogurt does he need to make 20 smoothies?
Answers
Answer:
40 smoothies
Step-by-step explanation:
if a 1/2 is one then times it by two
Answer:
40 smoothies
Step-by-step explanation:
The perimeter of a rectangular window is 43 feet. The width of the window is 5 ft more than the length. Find the width of the window
Answers
Answer:
13.25 ft
Step-by-step explanation:
Let w represent the width of the window. Then w-5 represents the length, and the perimeter is ...
P = 2(L+W)
43 = 2((w-5) +w) = 4w -10
53/4 = w = 13.25 . . . feet
The width of the window is 13.25 ft.
For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate limn→[infinity]Sn to obtain the value of the series or state that the series diverges.
1) ∑[infinity]k = 1(1/k + 1 - 1/k + 2).
2) ∑[infinity]k = 1(1/(k + 6)(k + 7).
Answers
Answer:
The following are the solution to the given points:
Step-by-step explanation:
Given value:
[tex]1) \sum ^{\infty}_{k = 1} \frac{1}{k+1} - \frac{1}{k+2}\\\\2) \sum ^{\infty}_{k = 1} \frac{1}{(k+6)(k+7)}[/tex]
Solve point 1 that is [tex]\sum ^{\infty}_{k = 1} \frac{1}{k+1} - \frac{1}{k+2}\\\\[/tex]:
when,
[tex]k= 1 \to s_1 = \frac{1}{1+1} - \frac{1}{1+2}\\\\[/tex]
[tex]= \frac{1}{2} - \frac{1}{3}\\\\[/tex]
[tex]k= 2 \to s_2 = \frac{1}{2+1} - \frac{1}{2+2}\\\\[/tex]
[tex]= \frac{1}{3} - \frac{1}{4}\\\\[/tex]
[tex]k= 3 \to s_3 = \frac{1}{3+1} - \frac{1}{3+2}\\\\[/tex]
[tex]= \frac{1}{4} - \frac{1}{5}\\\\[/tex]
[tex]k= n^ \to s_n = \frac{1}{n+1} - \frac{1}{n+2}\\\\[/tex]
Calculate the sum [tex](S=s_1+s_2+s_3+......+s_n)[/tex]
[tex]S=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+.....\frac{1}{n+1}-\frac{1}{n+2}\\\\[/tex]
[tex]=\frac{1}{2}-\frac{1}{5}+\frac{1}{n+1}-\frac{1}{n+2}\\\\[/tex]
When [tex]s_n \ \ dt_{n \to 0}[/tex]
[tex]=\frac{1}{2}-\frac{1}{5}+\frac{1}{0+1}-\frac{1}{0+2}\\\\=\frac{1}{2}-\frac{1}{5}+\frac{1}{1}-\frac{1}{2}\\\\= 1 -\frac{1}{5}\\\\= \frac{5-1}{5}\\\\= \frac{4}{5}\\\\[/tex]
[tex]\boxed{\text{In point 1:} \sum ^{\infty}_{k = 1} \frac{1}{k+1} - \frac{1}{k+2} =\frac{4}{5}}[/tex]
In point 2: [tex]\sum ^{\infty}_{k = 1} \frac{1}{(k+6)(k+7)}[/tex]
when,
[tex]k= 1 \to s_1 = \frac{1}{(1+6)(1+7)}\\\\[/tex]
[tex]= \frac{1}{7 \times 8}\\\\= \frac{1}{56}[/tex]
[tex]k= 2 \to s_1 = \frac{1}{(2+6)(2+7)}\\\\[/tex]
[tex]= \frac{1}{8 \times 9}\\\\= \frac{1}{72}[/tex]
[tex]k= 3 \to s_1 = \frac{1}{(3+6)(3+7)}\\\\[/tex]
[tex]= \frac{1}{9 \times 10} \\\\ = \frac{1}{90}\\\\[/tex]
[tex]k= n^ \to s_n = \frac{1}{(n+6)(n+7)}\\\\[/tex]
calculate the sum:[tex]S= s_1+s_2+s_3+s_n\\[/tex]
[tex]S= \frac{1}{56}+\frac{1}{72}+\frac{1}{90}....+\frac{1}{(n+6)(n+7)}\\\\[/tex]
when [tex]s_n \ \ dt_{n \to 0}[/tex]
[tex]S= \frac{1}{56}+\frac{1}{72}+\frac{1}{90}....+\frac{1}{(0+6)(0+7)}\\\\= \frac{1}{56}+\frac{1}{72}+\frac{1}{90}....+\frac{1}{6 \times 7}\\\\= \frac{1}{56}+\frac{1}{72}+\frac{1}{90}+\frac{1}{42}\\\\=\frac{45+35+28+60}{2520}\\\\=\frac{168}{2520}\\\\=0.066[/tex]
[tex]\boxed{\text{In point 2:} \sum ^{\infty}_{k = 1} \frac{1}{(n+6)(n+7)} = 0.066}[/tex]
The graph of f(x) = 0.5x is replaced by the graph of g(x) = 0.5x + k. If g(x) is obtained by shifting f(x) up by 5 units, then what is the value of k?
Answers
Answer:
k = 5
Step-by-step explanation:
The value of k is added to each y-coordinate, so represents the amount of up-shift. If the shift up is 5 units, then k=5.
Answer:
k = 5
Step-by-step explanation:
I got it right on the quiz
The flow-rate of your current sink faucet is 2.7 gallons per minute. If you purchase a fancy low-flow faucet for $294, the flow rate would decrease by 39%. Assuming you use your faucet for an average of 1,558 minutes per month and water costs $0.006 per gallon, compute the payback period (in months) of such a purchase.
Answers
Answer:
23
Step-by-step explanation:
How many proportional relationships are shown in the coordinate plane below?
Choose 1 answer:
(Choice A)
0
(Choice B)
1
(Choice C)
2
(Choice D)
3
Answers
Answer:
D. 3
Step-by-step explanation:
We can see from the graph that there are 3 lines. All 3 lines look linear, and if a line is linear, x and y are proportional. Therefore, if all 3 lines are linear, then we have 3 proportional relationships.
The translation rule is (x,y) (x+_,y+_)
Answers
Answer:
see below
Step-by-step explanation:
The question is not clear but I will provide info based on given
Translation of
(x, y) → (x + a, y+ b)means sliding by a points to left/right and by b points to up/down
If simplifying the following problem, what
would be the correct first and third
steps?
8 - 4 [5(2-2)] + 3
Answers
Answer: First step would be doing the () first. The third step would be 4 times 0. I think so, forgive me if I’m wrong.
Step-by-step explanation:
Kyanna has 4 pears, 6 apples, and 3 oranges. Write a ratio to show oranges to pears.
Answers
Answer:
3:4
Step-by-step explanation:
Use S = n^2 to find the sum of 1 + 3 + 5 + . . . + 701.
Answers
Answer:
123201
Step-by-step explanation:
This is an arithmetic sequence of common difference 2 and starting value 1,
So we can use the formula for an arithmetic sequence if we know what is the order of the last term.
Then first use the formula for the nth term to find "n":
[tex]a_n=a_1+(n-1)d[/tex]
where d = 2, first term = 1 and last term = 701
[tex]a_n=a_1+(n-1)d\\701=1+(n-1)*2\\700=(n-1)*2\\350=n-1\\n = 351[/tex]
Knowing this, we can estimate the partial sum:
[tex]S=n\,\frac{a_1+a_n}{2} \\S=351\,\frac{1+701}{2} \\S=351\,*\,351\\S = 123201[/tex]
You can multiply -2.2 by 6.2 to find the total variation in the water level.What decimal represents the total variation in water level Ericka observed?
Answers
Answer:
-13.64 millimeters
Step-by-step explanation:
just use a calculator ._.
Expand and simplify
(2x2 + 3)(4x + 5) - 8x(x - 2)
Answers
Robert is building model cars as well as model trains. The cars take 4 hours to build while the trains take 6 hours
to build. He is hoping to build as many as possible before the next model fair exhibition. If he has 100 free hours
until the fair, how many can he build? Write the inequality for this problem. Let cars be the domain and trains the
range.
Answers
10 cars and 10 trains in 100 hours
it approximately take him 10 hours to build one car and one train how 6+4=10 how many times can 10 go into 100 10 times hope this helps a little
There are 450 seats in a theatre. 48% of the seats are occupied. How many seats are not occupied?
Answers
Answer:
[tex] \boxed{ \bold{ \huge{ \boxed{ \sf{234}}}}}[/tex]
Step-by-step explanation:
Total seats in a theatre = 450
Finding the total occupied seats
[tex] \dashrightarrow{ \sf{48\% \: of \: 450}}[/tex]
[tex] \dashrightarrow{ \sf{ \frac{48}{100} \times 450}} [/tex]
[tex] \dashrightarrow{ \sf{216}}[/tex]
Finding the seats that are not occupied
[tex] \longrightarrow{ \sf{450 - 216 = 234}}[/tex]
Hope I helped!
Best regards! :D
