NotThatSmart

Alright, not too bad of a problem!

So, we just have to set up the slope equation to find what a+b is equal to because spoiler alert, this is a really easy problem once we write out the equation. 

The slope formula is where the slope of a line is equal to \(\frac{x2-x1}{y2 - y1}\) where (x2, y2) and (x1, y1) are points on that line. 

In this case, we have point 1 as (a,a^2) and point 2 as (b,b^2). 

Plugging these numbers into the formula, we have \(\frac{ b^2 - a^2}{b - a} = 2\). Now, there's on trick that helps us solve thus problem immediately. That is, Mr. Good Old Difference of Squares. 

We can write the numerator as \((b-a)(b+a)\), meaning we can just cancel out the b - a. 

\(\frac{(b -a) (b + a)}{b -a} = b + a = 2\)

Boom, this means b + a = 2, so our final answer is simply 2! Simple enough!

Thanks!

Alright! I geuss I can try to solve this question!

First, let's note that the only points where the lines intersect is when we have \(f(x) = g(x)\). 

This means that we only have to solve the equation \(x^3 + x^2 -3x + 5 = x^3 + 2x^2 \) to find the points where the lines intersect. 

Combining like terms and moving all the terms to one side, we get the equation \(x^2 + 3x - 5 = 0\).

We can't factor this polynomial with integers, so the next best thing to do is to complete the square and find x that way.

We set up the equation by moving 5 to the other side, getting us \(x^2 + 3x = 5\). Now we complete the square by adding 9/4 to both sides. 

\(x^2 + 3x + 9/4 = 5 + 9/4 \)

\((x + 3/2)^2 = 29/4\)

Now we just have to square root both sides to get us \(x+3/2= \pm \sqrt{29/4}\)

This means that \(x = \frac{\sqrt{29}-3}{2}, \frac{-\sqrt{29}-3}{2}\).

Subsituting these x values back into the g(x) and f(x) functions, we get \(y = \frac{-\sqrt{29}-3} 2, \frac{\sqrt{29}-3}{2}\)

So for the final answer, we have \((\frac{\sqrt{29}-3}{2}, \frac{-\sqrt{29}-3}{2}), (-\frac{\sqrt{29}-3}{2}, \frac{\sqrt{29}-3}{2})\)

Thanks! :) :) :)

We know that f(3) = 4, so we want the inside of f(2x/3) to equal 3. We notice that when we have x=9/2, then we have \(y=\frac{5+f(3)}{11} = \frac{5+4}{11}\). 

This is easily solvable, so we get y=9/11. 

This means that the point (9/2, 9/11) has to be on the line. 

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Let's first notice that both triangle BPQ and triangle CQP are both right triangles. 

In order for the sin of angle PAQ, we need to find the length of side PQ and PA and then use the ratio for the sin.

Using the pythagorean thereom, we can find that the length of PQ is \(PQ^2 = BP^2 + PC^2 PQ^2 = 4^2 + 1^2 PQ^2 = 17 PQ = \sqrt{17} \)

Use the Pythagorean Theorem, we can find the length of PA is \(PA^2 = CQ^2 + QP^2 PA^2 = 4^2 + \sqrt{17}^2 PA^2 = 33 PA = \sqrt{33}\)

Now, we simply take the ratio of the two numbers getting us \(\frac{\sqrt{17}}{\sqrt{33}} = 17/33\)

So the sin of PAQ is simply \(\sqrt{\frac{17}{33}}\)!

Thanks!

Alrgiht! Not too bad of a problem!

First of all, if the polygon has 3 sides then there are 3 angles, pretty simply in that term. It also means that the sides add up to 180!

Another thing to point out is the fact that because the 3 angles make an arithmatic sequence, the difference between the smallest and the middle angle is the same as the difference between the biggest and the middle angles. Since the total is 56, we just do 56/2 = 28, so the common difference for angles ig 28.

Now, let's write an equation. 

Let s be the smallest angle. We have: \(d +(d+28) + (d+56) = 180\)

Combining like terms gets us \(3d + 84 = 180\), meaning \(3d = 96\) so 

\(d = 96/3 = 32\)

So the smallest angle is 32 

Thanks!

You can't really tell the exact sidelength or an integer for this problem, but you can tell what the sudelength is in terms of the area or perimeter. 

First off, we know that the perimeter of a regular hexagon is 6 times the sidelength, meaning the area is \(3/2 * 6 = 9 \) times the area. 

So the sidelenghth of the hexagon is 1/9 the area and 1/6 the perimeter!

I hope I answered your question!

Thanks!

I think you are missing the graph that shows the graph of the exponential function. 

However, I can make a geuss to what the exponential function is using the two points :). 

Exponential functions are usually in the form of \(y=ax^b\) where a is the y intercept and b is the growth factor. 

With the two points you provided, f(0) = 1 is defintely the y intercept at point (0, 1), meaning the a value is 1. 

My best geuss at what the growth factor is would probably be that the y value doubles everytime x increases by 1, because that would make the most sense for a exponential function.

This would mean the b value or growth factor is 2. 

So, my best geuss without a graph of what f(x) is would probably be \(f(x) = 2^x\). 

I'm not sure if it's right, but it does satisfy the two points given, and that was all, so I hope it was a bit helpful!

Thanks!

So first off, let's write the function in standard form . Warning, this problem is quite straightforward, and you could probably geuss it...maybe

Distributing the 2 into the funcction gets us \(f(x) = -6x+4\)

DOMAIN: 

For the domain of this function, we just need to check whether or not there are any limits to what x can be. 

For example, in the function \(f(x) = \sqrt{2x+54}\), we must have the descriminant as greater or equal to 0, meaning there are restraints on the domain. 

In this function however, there are none, since x could be anything and it would not violate any concepts. 

Domain: All Real Numbers

RANGE:

Now, the range is limited usually because the domain is limited in someway. 

In this case, there are no limited to the domain, so there are no limits to the range. 

Range: All Real Numbers

Yep, both the range and domain are ALL REAL NUMBERS. 

In scientific notation:

Domain: \((-\infty, \infty)\)

Range: \((-\infty, \infty)\)

Thanks!