ilorty

Your answer was very great too! Although I am a bit confused on what the purple line means...

I did it differently, but Melody's answer is still very good. (Or maybe they are the same, and I just did not see...)

The first thing to do, as Melody did, is to set f(x) to y. This gives us:

y=(x^2 +14x+9)/(x^2+2x+3).

Now, we can multiply both sides by x^2+2x+3, and get:

y(x^2+2x+3)=x^2 +14x+9.

Now, we must solve for the discriminant:

x^2 + 14x + 9=yx^2+2yx+3y

x^2 + 14x + 9-(yx^2+2yx+3y)=0

(1-y)x^2 + (14-2y)x + 9-3y=0      <---- regrouping to get the coefficients

(14-y)^2-4(1-y)(9-3y)=196-56y+4y-(36-48y+12y^2)=-8y^2-8y+160      <--------discriminant formula

Now that we got our discriminant, we can say:

-8y^2-8y+160>=0   <------- or else it would be imaginary, which is no good

y^2+y-20 <= 0

(y+5)(y-4) <= 0

We see that for any number between -5 and 4, it works, so our range is [-5, 4]

:)

Thanks!

ikr, lol...it made my computer super laggy 

the beauty of math....

(i think it would be less scary better if i changed the colors....)

We can rewrite 4^(x-1) as 4^x/4. 4^x can be rewritten as (2^x)^2, which is 7^2, which is 49. So, the answer is 49/4, which is 12.25 in decimal form.

:)

I'm confused at what a unique polygon is......you mean a shape that follows the "unique shape theorem"? Then it would be B, because a square is a regular rectangle, since all sides are equal. 
 

(Also, can you please reply, telling me what a unique polygon is? Thanks!)

:)

To start, let's find what a and b are. Using the quadratic formula, we see that a=(23+sqrt341)/2, and b=(23-sqrt341)/2. Now, plug that into what we want: ((23+sqrt341)/2-2)((23-sqrt341)/2-2). Multiply it out, and you get: 5.

:)

For me, I can only see it in the preview, whether i log in or out...but it's not just the LaTeX that isn't working for me, but whenever I want to type a plain message, it ends up going into this small box instead of a normal box. (here is what i mean)

small box:

A continuation of what guest said:

First, we plug in -1 for x in the function f(x). That is -5+3=-2. Next, we plug in -2 into g(x). That is 4-2=2. 

:)