Mathematics College

Given that lim x→1 (5x − 3) = 2, illustrate definition 2 by finding values of δ that correspond to ε = 0.1, ε = 0.05, and ε = 0.01.

Answers

Answer 1
[tex]\displaystyle\lim_{x\to1}(5x-3)=2[/tex]

is to say that for any [tex]\varepsilon>0[/tex], we can find [tex]\delta>0[/tex] such that [tex]0<|x-1|<\delta[/tex] implies [tex]|(5x-3)-2|<\varepsilon[/tex].

To that end, we have

[tex]|(5x-3)-2|=|5x-5|=5|x-1|<\varepsilon\implies|x-1|<\dfrac\varepsilon5=\delta[/tex]

which means that if [tex]\varepsilon=0.1[/tex] is given, then we can choose [tex]\delta=\dfrac{0.1}5=0.02[/tex]; if [tex]\varepsilon=0.05[/tex], then [tex]\delta=\dfrac{0.05}5=0.01[/tex]; if [tex]\varepsilon=0.01[/tex], then [tex]\delta=\dfrac{0.01}5=0.002[/tex].
Answer 2
Final answer:

To illustrate the definition of a limit, we need to find values of δ for different values of ε. We can solve inequalities involving the expression (5x − 3) - 2 to find suitable values of δ for ε = 0.1, ε = 0.05, and ε = 0.01.

Explanation:

Limits are an important concept in calculus. In this case, we are given that the limit as x approaches 1 of the expression (5x − 3) is 2. To illustrate the definition, we need to find values of δ that correspond to different values of ε. We can start by setting ε to 0.1 and then solve for δ.

When ε = 0.1, we want to find δ such that |(5x − 3) - 2| < 0.1 whenever 0 < |x - 1| < δ. We can solve this inequality by manipulating it and simplifying it to find a suitable value of δ. Similarly, we can find values of δ corresponding to ε = 0.05 and ε = 0.01 by solving the inequalities |(5x − 3) - 2| < 0.05 and |(5x − 3) - 2| < 0.01, respectively.

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Related Questions

What is 1/4 of 268 ?

Answers

The answer to your question is not very complicated, you just had to divide 268 by 4 and you'd get 67
Answer: 67
Final answer:

To find 1/4 of 268, you divide 268 by 4. The result of this operation is 67.

Explanation:

Calculating 1/4 of a number involves a simple mathematical operation known as division. In this case, we want to find out what is 1/4 of 268. To do this, you simply divide 268 by 4.

The calculation is as follows:

Divide the number 268 by 4.268 ÷ 4 equals 67.

This means that 1/4 of 268 is 67.

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53 ℃ below zero degrees

Answers

I don't really understand this...
your answer is -53 degrees celsius

hope this helps

Devaughn is 13 years older than Sydney. The sum of their ages is 77 . What is Sydney's age?

Answers

77-13=64....64÷2=32 the age should be 32

other form to write 600,000+80,000+10

Answers

six hundred eighty thousand and ten

hope this helps


or 680,010

how to solve this linear equation step by step please

Answers

[tex]\bf \cfrac{3}{c^2-4c}-\cfrac{9}{2c^2+3c}=\cfrac{2}{2c^2-5c-12}\\\\\\ \begin{cases} c^2-4c=&c(c-4)\\ 2c^2+3c=&c(2c+3)\\\\ 2c^2-5c-12= &(2c+3)(c-4) \end{cases}\\\\ -------------------------------\\\\ \textit{therefore, our LCD for the \underline{left-side} will have to be }\underline{c(c-4)(2c+3)}\\\\ -------------------------------\\\\[/tex]

[tex]\bf \cfrac{3}{c(c-4)}-\cfrac{9}{c(2c+3)}=\cfrac{2}{(2c+3)(c-4)} \\\\\\ \cfrac{3(2c+3)-9(c-4)}{c(c-4)(2c+3)}=\cfrac{2}{(2c+3)(c-4)} \\\\\\ \cfrac{6c+9-(9c-36)}{c(c-4)(2c+3)}=\cfrac{2}{(2c+3)(c-4)} \\\\\\ \cfrac{6c+9-9c+36}{1}=\cfrac{2[c(c-4)(2c+3)]}{(2c+3)(c-4)} \\\\\\ -3c+45=2c\implies 45=5c\implies \cfrac{45}{5}=c\implies 9=c[/tex]

From 1980 to 1990, the consumer price index (CPI) increased from 82.4 to 130.7. If a bottle of dish soap cost $0.74 in 1980 and the price of dish soap increased at the same rate as the CPI from 1980 to 1990, by approximately how much did the price of a bottle of dish soap increase from 1980 to 1990?

Answers

Using proportions, it is found that the price of a bottle of dish soap increased $0.43 from 1980 to 1990.

What is a proportion?

A proportion is a fraction of a total amount, and the measures are related using a rule of three.

The percent increase, as a proportion, of the CPI is given by:

P = 130.7/82.4 = 1.5862.

Hence, for the price of the bottle:

Pb = $0.74 x 1.5862 = $1.17.

The difference is:

1.17 - 0.74 = $0.43.

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Answer: $0.43

Step-by-step explanation:

The cool dude above me already explained it

There are two aircraft carriers, A and B, and the carrier A is longer in length the carrier B. The total length of these two carriers is 4178 feet, while the difference of their lengths is only 14 feet.

Answers

Answer:

Length of carrier A = 2096 feet Length of carrier B=2082 feet.

Step-by-step explanation:

Given , two air craft carriers are A and B . The carrier A is longer than the carrier B in length.

Total length of two carriers =4178 feet

Difference of two carriers = 14 feet

Let

Length of carrier A =x  feet

And

Legth of carrier B =y feet

According to question  

The I equation is

x+y=4178

The II equation is

x-y= 14

Substitution method:  In this method step I :  we  find value of x or y from the equation I

Step II:  put the value of x or y in II equation then we get the value of another  varaiable x or y  

Step : Again put the obatined value of variable x or y  from  step II   in equation I then we get value of other variable x or y .

By  using substitution method ,

we find value of x from equation I

x= 4178-y

Now, put the value of x in equation II then we get

4178-y-y=14

4178-2y=14

4178-14=2y

4164=2y

[tex]y=\frac{4164}{2}[/tex]

y= 2082

Again , put the value of y in equation I

2082 +x=4178

x= 4178-2082

x=2096

Hence, the length of carrier A = 2096 feet

The length of carrier B= 2082 feet

Final answer:

To solve the problem, we set up two equations based on the given total length and the length difference of the two aircraft carriers. By solving these equations, we find that carrier A is 2096 feet long and carrier B is 2082 feet long.

Explanation:

The student's question involves solving a system of linear equations to find the lengths of two aircraft carriers, A and B. Given that the total length is 4178 feet and the length difference is 14 feet, we can set up the following equations:

Equation 1: A + B = 4178

Equation 2: A - B = 14

To solve for A and B, we can add the two equations together to eliminate B.

This gives us 2A = 4192, hence A = 2096 feet.

Substituting A back into one of the equations, we find B = 2082 feet.

Therefore, carrier A is 2096 feet long and carrier B is 2082 feet long.

If you earn $3500 per month and you expect your earnings to increase by 2.3% per year, how much do you think you will be making in 10 years? (Express your answer rounded correctly to the nearest cent!)

Answers

[tex]\bf \qquad \textit{Amount for Exponential Growth}\\\\ A=I(1 + r)^t\qquad \begin{cases} A=\textit{accumulated amount}\\ I=\textit{initial amount}\to &3500\\ r=rate\to 2.3\%\to \frac{2.3}{100}\to &0.023\\ t=\textit{elapsed time}\to &10\\ \end{cases} \\\\\\ A=3500(1+0.023)^{10}[/tex]

Final answer:

Your expected monthly earnings in 10 years, with a 2.3% annual increase, would be $4424.10, rounded to the nearest cent.

Explanation:

To calculate your expected monthly earnings in 10 years, taking into account a 2.3% annual increase, we will use the formula for compound interest: P(1 + r)^n, where:

P is the principal amount (your current earnings)

r is the annual raise rate

n is the number of years

Your current monthly earnings are $3500. The annual raise rate is 2.3%, so r is 0.023 when expressed as a decimal. The number of years, n, is 10.

Plugging the values into the formula, we get:

Earnings in 10 years = $3500 * (1 + 0.023)^10

Calculating the result, we find that your expected monthly earnings in 10 years are:

$4424.10

A dozen eggs cost $1.10 in Dover.  In Ensley, the eggs cost 10% more than in Dover.  Find the price of a dozen eggs in Ensley.

Answers

1.10 multiplied by .10 = .11
1.10 + .11 = 1.21

In Ensley, the price of a dozen eggs costs $1.21.

Hope this helps.

Final answer:

The price of a dozen eggs in Ensley is $1.21.

Explanation:

To find the price of a dozen eggs in Ensley, we need to consider that the eggs in Ensley cost 10% more than in Dover. If a dozen eggs in Dover cost $1.10, we can calculate the 10% increase by multiplying $1.10 by 1.10:

$1.10 x 1.10 = $1.21

Therefore, a dozen eggs in Ensley cost $1.21.

Explain the place value relationship when the same two digits are next to each other in a multi-digit number.

Answers

Take one value as an example, 5664 where there are two digits of '6' next to each other

Partitioning this value, 
5664 = 5000 + 600 + 60 + 4

Notice that one digit of 6 worth 100 and the other worth 10

Take another example, 78869 where there are two digits of '8'

Partitioning this value, we have
78869 = 70000 + 8000 + 800 + 60 + 9

Notice that one digit of '8' worth 8000 and the other worth 800

One other characteristic that we can tell from both example, is that when two same digits are placed next to each other in a value, one digit's value is always worth ten times than the other digit.

for example, the value 600 and 60 from the first example, 600 = 10 × 60

What's the correct answer for this?

Answers

[tex]\bf \qquad \qquad \textit{inverse proportional variation}\\\\ \textit{\underline{y} varies inversely with \underline{x}}\qquad \qquad y=\cfrac{k}{x}\impliedby \begin{array}{llll} k=constant\ of\\ \qquad variation \end{array}\\\\ -------------------------------\\\\[/tex]

[tex]\bf \textit{\underline{N} is inversely proportional to \underline{A}}\qquad N=\cfrac{k}{A} \\\\\\ \textit{we also know that } \begin{cases} A=3\\ N=16 \end{cases}\implies 16=\cfrac{k}{3}\implies 16\cdot 3=k \\\\\\ 48=k\qquad thus\qquad \boxed{N=\cfrac{48}{A}}\\\\ -------------------------------\\\\ \textit{if \underline{A} is 4, what is \underline{N}?}\qquad N=\cfrac{48}{4}[/tex]

Determine whether each of the functions log(n + 1) and log(n2 + 1) is o(log n)

Answers

Assuming the order required is as n-> inf.

As n->inf, o(log(n+1)) -> o(log(n)) since the 1 is insignificant compared with n.

We can similarly drop the "1" as n-> inf, the expression becomes log(n^2+1) ->
log(n^2)=2log(n)  which is still o(log(n)).

So yes, both are o(log(n)).

Note: you may have more offers of answers if you post similar questions in the computer and technology section.

Both functions log(n + 1) and log(n^2 + 1) are not o(log n). For large values of n, log(n + 1) is approximately equal to log n, and log(n^2 + 1) behaves like 2*log n, none of which grows strictly slower than log n.

To determine whether the functions log(n + 1) and log(n2 + 1) are o(log n), we'll use the definition of little-o notation. A function f(n) is said to be o(g(n)) if for any positive constant c > 0, there exists a threshold n0 such that for all n > n0, f(n) < c * g(n). In other words, f(n) grows slower than any constant multiple of g(n) as n approaches infinity.

Analysis for log(n + 1)

When n is very large, the +1 becomes negligible, and log(n + 1) behaves similarly to log n. Therefore,

log(n + 1) ≈ log n when n is large.

This implies that log(n + 1) is not o(log n) because it does not grow strictly slower than any multiple of log n.

Analysis for log(n2 + 1)

Using logarithm properties:

log(n2 + 1) < log(n2 + n2) = log(2n2) = log 2 + 2*log n.

As n grows, the constant term (log 2) becomes insignificant, and the function approaches the behavior of 2*log n. However, since there is a multiple of log n (which is 2*log n), this still means that log(n2 + 1) is not o(log n) because it grows faster, not slower, than log n.

The building of Jim's Hardware is assessed at $109,000. The tax rate is $86.95 per $1,000 of assessed valuation. The tax due is A. $8,695.45. B. $947.75. C. $9,477.55. D. $8,659.54. E. 94,698.23.

Answers

The answer is C, $9,477.55

109,000 / 1,000 = 109

109 x 86.95 = 9,477.55

If the rate of inflation is 3.7% per year, the future price pt (in dollars) of a certain item can be modeled by the following exponential function, where t is the number of years from today. =pt400( 1.037)t Find the current price of the item and the price 8 years from today. Round your answers to the nearest dollar as necessary.

Answers

well, for the current price today, let's see, not even one day has passed, to we don't have a "t" value that's greater than 0, so since today 0years and 0days and 0seconds have passed, t =0

[tex]\bf p(t)=400(1.037)^0\implies p(t)=400\cdot 1\implies \boxed{p(t)=400}[/tex]

now, 8 years from now, well, 8 years would had passed by then, t = 8

[tex]\bf p(t)=400(1.037)^8\implies p(t)\approx 400\cdot 1.3373037\implies \boxed{p(t)\approx 535}[/tex]
P (t)=400 (1.037)^t
The current price is 400

The price 8 years from today is
P (8)=400×(1.037)^(8)=534.9 round your answer to get 535

What is the step by step process of solving for the GCF from a list of terms.
Ex. (see picture)

Answers

[tex]\bf 18a^5b^8c^4\quad \begin{cases} 18=\underline{2\cdot 3}\cdot 3\\ a^5=\underline{aaaaa}\\ b^8=\underline{b} bbbbbbb\\ c^4=cccc \end{cases}\qquad 12a^6b\quad \begin{cases} 12=2\cdot \underline{2\cdot 3}\\ a^6=\underline{aaaaa} a\\ b=\underline{b} \end{cases} \\\\\\ \textit{notice the underlined factors, those are common to both terms} \\\\\\ 2\cdot 3\cdot aaaaa\cdot b\implies 6a^5b\impliedby GCF[/tex]

A and B are mutually exclusive events. P(A) = 1/3 and P(B) = 1/2. What is the P(A or B)?

Answers

When you have mutually exclusive events and what to find the probability of one or another, add the probabilities.
1/3+1/2=5/6
Final answer: 5/6
P(A or B) = 1/2 + 1/3 = 3/6 + 2/6 = 5/6

answer
P(A or B) = 5/6

A single batch of cookies requires two cups of sugar and three cups of flour. if more than one batch or a partial batch was made, the unit rate of sugar to flour would remain the same. match the correct amount of sugar to the correct amount of flour to keep the same unit rate of sugar and flour in a single batch. 11.5 cups of flour 4.5 cups of sugar 6.75 cups of flour 12 cups of flour 5 cups of sugar 11.25 cups of flour 7.5 cups of sugar 7.5 cups of flour 8 cups of sugar 4 cups of sugar

Answers

The ratio of flour to sugar is 3:2, whatever the unit is.

First write down the amounts of flour (in cups)
11.5 
6.75
12
11.25
7.5
then write down two-third of the values of flour on the right to represent the number of cups of sugar
11.5    7.67
6.75   4.5
12      8
11.25 7.5
7.5     5

Then check if all the amounts of sugar are on the list.
It seems that 6 cups of flour (to match 4 cups of sugar) and 7.67 cups of sugar (to match 11.5 cups of flour) are missing.

Please check for typos.

Answer:11.5

Step-by-step explanation: 11.5

Manuel is choosing a 3 -letter password from the letters A, B, C, D, and E. The password cannot have the same letter repeated in it. How many such passwords are possible?

Answers

Here they are... ABC ABD ABE ACB ACD ACE ADB ADC ADE.. Just do the same thing with the others... Ex. BCD BDE BCE BCA BDA BDC BDE etc. 

Use the half-angle identities to find the exact value of cos 15 degrees.

This is what I have so far: 

cos15 degrees = cos1/2(30 degree) = sqrt (1+cos30)/2 = sqrt (1+ sqrt3/2)/ 2

But.. I don't understand how the cos30 turns into sqrt 3/2??

Answers

The question is: how does cos(30) = [tex] \frac{ \sqrt{3} }{2} [/tex]


Let's start by considering the equilateral triangle ABC in the diagram below. An equilateral triangle has equal size of angles, 60°, 60°, and 60°. Let's say the length of each side to be 2 units.

If we split this triangle into two congruent right-angled triangle, we will have the measurement of the perpendicular height to be √3 by Pythagoras theorem.

By trigonometry ratio we then have [tex]cos(30) = \frac{ \sqrt{3} }{2} [/tex]

This ratio will always work for any right angle triangle with angles 90°, 60° and 30°

Going back to the original question:

cos(15°) = cos(30°) which is cos([tex] \frac{30}{2} [/tex])
let θ be 30° then cos(15°)=cos(θ/2)
by the trigonometry formula for half angle identity
cos(θ/2) = √(1+cosθ)/2
cos(15) = √(1+cos(30))/2
cos(15) = √(1+[tex] \frac{ \sqrt{3}} {2} [/tex]) / 2
cos(15) = [tex] \frac{ \sqrt{2+ \sqrt{3} } }{2} [/tex]


The cosine value of cos(15) is [tex]\sqrt[/tex](1 + [tex]\sqrt[/tex]3/2)/2

The trigonometry identity of half angles is given as:

cos([tex]\theta[/tex]/2) = [tex]\sqrt{[/tex](1 + cos([tex]\theta[/tex]))/2

Substitute 30 for [tex]\theta[/tex]

So, the equation becomes

cos(30/2) = [tex]\sqrt[/tex](1 + cos(30))/2

In trigonometry, we have:

cos(30) = [tex]\sqrt[/tex]3/2

So, we have:

cos(30/2) = [tex]\sqrt[/tex](1 + [tex]\sqrt[/tex]3/2)/2

Divide 30 by 2

cos(15) = [tex]\sqrt[/tex](1 + [tex]\sqrt[/tex]3/2)/2

Hence, the cosine value of cos(15) is [tex]\sqrt[/tex](1 + [tex]\sqrt[/tex]3/2)/2

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Mark deposited $9,000 into two saving accounts bearing simple interest. One of the accounts has an interest rate of 8% while the other rate is 7%. If the total interest earned after one year is $700, find the amount deposited into each of the accounts

Answers

0.07x+0.08 (9000-x)=700
Solve for x
0.07x+720-0.08x=700
0.07x-0.08x=700-720
-0.01x=-20
X=20÷0.01
X=2,000 invested at 7%

9000-2000=7000 invested at 8%

Malia is observing the velocity of a cyclist at different times. After two hours, the velocity of the cyclist is 15 km/h. After five hours, the velocity of the cyclist is 12 km/h.
Part A: Write an equation in two variables in the standard form that can be used to describe the velocity of the cyclist at different times. Show your work and define the variables used. (5 points)
Part B: How can you graph the equations obtained in Part A for the first 12 hours? (5 points)

Answers

We must to remember what is the standard form of equation of straight line. It looks like that:
Ax+By+C=0.  A,B and C it's coefcients; x and y - it's variables.A and B must not be equal to zero at same time.This condition can be writen like this: (A^2+B^2) not equal to zero.
We know that x=2 ("after two hours") and y=15 ("velocity of the cyclist is 15 km/h")
AND
x=5 ("after five hours") and y=12 ("velocity of the cyclist is 12 km/h").
Lets look at standart form and give the values on variables.1. 2A+15B+C=0

2. 5A+12B+C=0and it is system.let's subtract the bottom equation from the top  -3A+3B=03A=3BA=B.So we can choose any value for A and B, just A=B.I choose for A=B=1.Our equation takes the formx+y+C=0
x+y=-C
and if we giving the value to variables, we can find coefficients "C".
5+12=-CC=-17.
Standart form is:x+y-17=0. ---- answer for Part A.
and the graph of this function is a straight line.For find the answer to Part B, we need to prolong our straight line into the perpendicular vertical line wich contain x=12.
In the point of the intersection of the lines, you must to turn at right angle to axis OY and to come at the value in the axis OY, it's will be the velocity after 12 hours.  

Round 241,639 to the nearest thousand

Answers

Hi!

The thousandths place is the fourth digit to the left of the decimal.

241,639

Look to the right of the 1.
Since 6 is greater than 4, we round up. 

242,000

The answer is 242,000

Hope this helps! :)

Two perpendicular lines intersect at the origin. If the slope of the first line is .5, what is the equation of the second line?. .

Answers

The origin is at the point (0,0). Therefore, the y-intercept will be 0. For two points to be perpendicular, the slopes must be opposite reciprocals of each other. .5 can be seen as 1/2, so the reciprocal is 2, and it is positive, making that 2 negative. Your equation would be y = -2x. 

Round 65.85 to the nearest whole number

Answers

66.

Since the value after the decimal place is above 5, it rounds to 66 and not 65

What is the value of b2 - 4ac for the following equation? x(x + 8) = 9 28 64 100

Answers

Answer:

C) 100

Step-by-step explanation:

The given equation is x(x + 8) = 9

Distributing x insides, we get

x^2 + 8x = 9

Now set the equation equal to zero.

x^2 + 8x - 9 =0

Here a = 1, b = 8, and c = -9

b^2 - 4ac = (8)^2 - 4*1*-9

= 64 + 36

= 100

Therefore, b^2 - 4ac = 100.

Answer: C) 100

Hope this will helpful.

Thank you.

Answer:

C. 100

Step-by-step explanation:

Find the value of each determinant

Answers

now, recall that, to get the determinant of a 3x3 or higher matrix, you pick a column or row to get the cofactors, also recall that the cofactors need to go witht the checkerboard pattern, so   [tex]\bf \begin{bmatrix} +&-&+&-&+&...\\ -&+&-&+&...\\ +&-&+&-&+&... \end{bmatrix}[/tex]  and so on.

so, let's use the last column, 2,0,4, because it has a zero there, and we can use that as a cofactor to simply get a 0 as a product with the minor.

[tex]\bf \begin{bmatrix} 4&-1&\boxed{2}\\ 6&-1&\boxed{0}\\ 1&-3&\boxed{4} \end{bmatrix}\impliedby \textit{let's use the 3rd column for our cofactors}\\\\ -------------------------------\\\\ \begin{bmatrix} &&\boxed{2}\\ 6&-1&\\ 1&-3& \end{bmatrix}\implies +2 \begin{bmatrix} 6&-1\\1&-3 \end{bmatrix}\implies +2[(-18)-(-1)] \\\\\\ +2(-17)\implies -34\\\\ -------------------------------\\\\[/tex]

[tex]\bf \begin{bmatrix} 4&-1&\\ &&\boxed{0}\\ 1&-3& \end{bmatrix}\implies -0 \begin{bmatrix} 4&-1\\1&-3 \end{bmatrix}\implies -0[(-12)-(-1)]\implies 0\\\\ -------------------------------\\\\ \begin{bmatrix} 4&-1&\\ 6&-1&\\ &&\boxed{4} \end{bmatrix}\implies +4 \begin{bmatrix} 4&-1\\6&-1 \end{bmatrix}\implies +4[(-4)-(-6)] \\\\\\ +4[2]\implies 8\\\\ -------------------------------\\\\[/tex]

so our determinant comes down to the sum of those three products... so -34 -0 + 8.

and surely you know how much that is.
Final answer:

The question is about finding the value of a determinant for matrices, a fundamental concept in Mathematics, especially linear algebra. For 2 × 2 matrices, the determinant calculation is straightforward and essential for various applications.

Explanation:

The subject of this question is clearly Mathematics, specifically it pertains to linear algebra and the concept of determinants. The task involves finding the value of a determinant for a given matrix. For a 2 × 2 matrix, the determinant is found using a simple formula: if the matrix is given by

\[\begin{pmatrix} a & b \\ c & d \end{pmatrix}\]

then the determinant is calculated as \(ad - bc\). Additionally, the determinant provides important information about the matrix, such as whether the matrix is invertible and the product of its eigenvalues.

To understand determinants for larger matrices, a recursive approach is often used, breaking down the determinant into smaller matrices until 2 × 2 matrices are reached, where the simple formula can be applied. Also of note is that the determinant of the product of two matrices is equal to the product of their determinants (det(AB) = det(A)det(B)).

This fundamental concept is crucial for many applications in mathematics, including solving systems of linear equations, finding eigenvalues, and understanding linear transformations.

write 103,727,495 in word form and expanded

Answers

one hundred three million seven hundred twenty seven thousand four hundred ninety five. 100,000,000+3,000,000+700,000+20,000+ 7,000+400+90+5
Word Form:
One hundred three million, seven hundred twenty seven thousand, four hundred ninety five

Expanded Form:
100,000,000 + 3,000,000 + 700,000 + 20,000 + 7,000 + 400 + 90 + 5

Hope this helped

7^-2 without exponent

Answers

[tex]\bf \left.\qquad \qquad \right.\textit{negative exponents}\\\\ a^{-{ n}} \implies \cfrac{1}{a^{ n}} \qquad \qquad \cfrac{1}{a^{ n}}\implies a^{-{ n}} \qquad \qquad a^{{{ n}}}\implies \cfrac{1}{a^{-{{ n}}}}\\\\ -------------------------------\\\\ 7^{-2}\implies \cfrac{1}{7^2}\implies \cfrac{1}{49}[/tex]
in any negative exponent problem all you do is flip the number(s) and or variable(s).
Ex: (7x)^-1 ----> 1 over 7x or 1/7x
Ex: (5/8x)^-1 ------> 8x over 5 or 8x/5

The spending limit on John’s credit card is given by the function f(x)=15,000+1.5x , where x is his monthly income. f^-1x . The variable x represents in the inverse function. If John's spending limit is $60,000, his monthly income is .

Answers

1.
"The spending limit on John’s credit card is given by the function f(x)=15,000+1.5x"

means that if the monthly income of John is $ 5,000 ,he can spend at most 

f(5,000)=15,000+1.5*5,000=15,000+ 7,500=22, 500 (dollars)

Or for example

if Johns monthly income is $8,000, then he can spend at most

 f(8,000)=15,000+1.5*8,000=15,000+ 12,000=27,000 (dollars)


2.
Now, assume that the maximum amount that John can spend is y.

Then, y=15,000+1.5x

we can express x, the monthly income, in terms of y by isolating x:

y=15,000+1.5x

1.5x = y-15,000


[tex]x= \frac{y-15,000}{1.5} [/tex]

thus, in functional notation, x, the monthly income, is a function , say g, of variable y, the max amount:

[tex]x=g(y)=\frac{y-15,000}{1.5} [/tex]

since we generally use the letter x for the variable of a function, we write g again as:

[tex]g(x)=\frac{x-15,000}{1.5} [/tex]

for example :

[tex]g(50,000)=\frac{50,000-15,000}{1.5}= \frac{35,000}{1.5}= 23,333 [/tex]

tells us that if the maximum amount that John can spend is 50,000 $, then his monthly income is 23,333 $.

3.

If John's limit is $60,000, his monthly income is 

[tex]g(60,000)=\frac{60,000-15,000}{1.5}= \frac{45,000}{1.5}=30,000 [/tex]
 dollars.


Answer: $ 30,000

Remark: g is called the inverse function of f, since it undoes what f does.

instead of g(x), we could use the notation [tex]f ^{-1}(x) [/tex]



A rectangular yard measuring 26ft by 40ft is bordered (and surrounded) by a fence. Inside, a walk that is 2ft wide goes all the way along the fence. Find the area of this walk. Be sure to include the correct unit in your answer.

Answers

In order to find the area of the walk, first you must find the area of the yard, then the area of the walk and yard combined. 

The area of the yard (minus the walk) is simple to find. Just multiply 26 and 40, and you'd get 1,040 ft squared. 

The area of the yard including the walk is a bit more complicated. You have to include add measurement of the walk to the measurements of the length and width of the yard. However, you must add it twice, considering it goes all the way around the yard. So, since the walk is 2 feet wide, you add it (twice) to 26 (26 + 4 = 30) and twice to 40 (40 + 4 = 44). Now, multiply the two new measurements. You now have 1,320 feet squared.

So now you have the combined area of the walk and yard, and the area of the yard without the walk. To find the area of the walk, you just subtract the area of the yard from the entire measurement of both. 1,320 square feet - 1,040 square feet is 280 square feet.

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