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3 years ago

Does the graph shown below represent y as a linear function of x?

Mathematics

1 answer:

marshall27 [118]3 years ago

5 0

Answer: No, it is not a linear function

Explanation:

We can see that it's a function because it passes the vertical line test, but it's not linear. To be linear, it needs to be one single straight line only (instead of composed of multiple pieces of different straight lines). This graph is considered a piecewise function.

So if your teacher asked "is this a function?" then the answer would be "yes". However, the "linear" portion is what makes the final answer "no".

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<img src="https://tex.z-dn.net/?f=%24a%2Ba%20r%2Ba%20r%5E%7B2%7D%2B%5Cldots%20%5Cinfty%3D15%24%24a%5E%7B2%7D%2B%28a%20r%29%5E%7B

riadik2000 [5.3K]

Let

$S_n = \displaystyle \sum_{k=0}^n r^k = 1 + r + r^2 + \cdots + r^n$

where we assume |r| < 1. Multiplying on both sides by r gives

$r S_n = \displaystyle \sum_{k=0}^n r^{k+1} = r + r^2 + r^3 + \cdots + r^{n+1}$

and subtracting this from $S_n$ gives

$(1 - r) S_n = 1 - r^{n+1} \implies S_n = \dfrac{1 - r^{n+1}}{1 - r}$

As n → ∞, the exponential term will converge to 0, and the partial sums $S_n$ will converge to

$\displaystyle \lim_{n\to\infty} S_n = \dfrac1{1-r}$

Now, we're given

$a + ar + ar^2 + \cdots = 15 \implies 1 + r + r^2 + \cdots = \dfrac{15}a$

$a^2 + a^2r^2 + a^2r^4 + \cdots = 150 \implies 1 + r^2 + r^4 + \cdots = \dfrac{150}{a^2}$

We must have |r| < 1 since both sums converge, so

$\dfrac{15}a = \dfrac1{1-r}$

$\dfrac{150}{a^2} = \dfrac1{1-r^2}$

Solving for r by substitution, we have

$\dfrac{15}a = \dfrac1{1-r} \implies a = 15(1-r)$

$\dfrac{150}{225(1-r)^2} = \dfrac1{1-r^2}$

Recalling the difference of squares identity, we have

$\dfrac2{3(1-r)^2} = \dfrac1{(1-r)(1+r)}$

We've already confirmed r ≠ 1, so we can simplify this to

$\dfrac2{3(1-r)} = \dfrac1{1+r} \implies \dfrac{1-r}{1+r} = \dfrac23 \implies r = \dfrac15$

It follows that

$\dfrac a{1-r} = \dfrac a{1-\frac15} = 15 \implies a = 12$

and so the sum we want is

$ar^3 + ar^4 + ar^6 + \cdots = 15 - a - ar - ar^2 = \boxed{\dfrac3{25}}$

which doesn't appear to be either of the given answer choices. Are you sure there isn't a typo somewhere?

7 0

2 years ago

Does anyone know the answer? (I need help ASAP please)

natima [27]

Answer:

D. (-10, -3), (-3, -10)

Step-by-step explanation:

For a question like this, it is easiest to check the offered answers in the given equations.

The first equation requires the sum of the x- and y-values to be -13. Adding two numbers is pretty easy, so you can rapidly determine that choice D is the only reasonable choice.

_____

Comment on the process

Answering multiple-choice questions is as much about test-taking skill as it is about math skill. First, you eliminate choices that don't answer the question.

Then you see if there is a way to identify the "correct" answer from the remaining "reasonable" answers. Often, that requires you only solve part of the problem, or you check for consistency between parts of the answer. (Here, if (a, b) is an answer, then (b, a) will be the other answer. Again, only choice D has that characteristic.)

You can see if the answer choices satisfy the details of the problem requirements.

Finally, as a last resort, you actually work the problem to determine your own answer to the question. (Here, you can substitute for y: x^2 +(-13-x)^2 = 109, then solve the quadratic, 2x^2 +26x +60 = 0 to find x=-3 or -10.) When you're done with this, you also need to check your answer against the above criteria.

5 0

3 years ago

Make x the subject of the formula y=7x+4

Sergio039 [100]

Answer:

x = (y - 4)/7

Step-by-step explanation:

Subtract 4 from both sides to get:

y - 4 = 7x

Then, divide both sides by 7 to get:

(y - 4)/7 = x

:)>

7 0

3 years ago

Read 2 more answers

Is it true that x= -5 ? -2(-4x-1)-x+3=−30 Justify your answer.

grandymaker [24]

Answer:

Step-by-step explanation:

here you go mate

step 1

-2(-4x-1)-x+3=−30 equation/Question

step 2

-2(-4x-1)-x+3=−30 distribute

7x+5=-30

step 3

7x+5=-30 subtract 5

7x=-35

step 4

$\frac{7x}{7} =\frac{-35}{7}$ divide

answer

x=-5

6 0

3 years ago

Read 2 more answers

Write the rate as a unit rate. 44 points in 4 quarters

Margaret [11]

Answer:

44 p points in 4 quarters = llets per Qtr.

3 0

3 years ago