MathProblemSolver101

\(\begin{align*} \frac{1}{1+\frac{1}{\sqrt{3}+2}} &= \frac{1}{\frac{3+\sqrt{3}}{\sqrt{3}+2}} \\ &= \frac{\sqrt{3}+2}{3+\sqrt{3}} \\ &= \boxed{\frac{3+\sqrt{3}}{6}} \end{align*}\)

Solution:

$f(x) = \frac{x - \sqrt{3}}{x\sqrt{3} + 1}$

\(\begin{align*} f^2(x) &= \frac{\frac{x-\sqrt{3}}{x\sqrt{3}+1}-\sqrt{3}}{\sqrt{3}\left(\frac{x-\sqrt{3}}{x\sqrt{3}+1}\right)+1} \\ &= \frac{\frac{x-\sqrt{3}}{\sqrt{3}x+1}-\sqrt{3}}{\frac{\sqrt{3}\left(x-\sqrt{3}\right)}{\sqrt{3}x+1}+1} \\ &= \frac{\frac{-2x-2\sqrt{3}}{\sqrt{3}x+1}}{\frac{\sqrt{3}\left(x-\sqrt{3}\right)}{\sqrt{3}x+1}+1} \\ &= \frac{-2x-2\sqrt{3}}{\left(x\sqrt{3}+1\right)\left(\frac{\left(x-\sqrt{3}\right)\sqrt{3}}{x\sqrt{3}+1}+1\right)} \\ &= \frac{-2x-2\sqrt{3}}{\frac{2\sqrt{3}x-2}{\sqrt{3}x+1}\left(\sqrt{3}x+1\right)} \\ &= \frac{-2x-2\sqrt{3}}{2\sqrt{3}x-2} \\ &= -\frac{2\left(x+\sqrt{3}\right)}{2\left(\sqrt{3}x-1\right)} \\ &= -\frac{x+\sqrt{3}}{\sqrt{3}x-1} \end{align*}\)

\(\begin{align*} f^3(x) &= f \left(-\frac{x+\sqrt{3}}{\sqrt{3}x-1} \right) \\ &= -\frac{\frac{x-\sqrt{3}}{x\sqrt{3}+1}+\sqrt{3}}{\sqrt{3}\left(\frac{x-\sqrt{3}}{x\sqrt{3}+1}\right)-1} \\ &= \frac{\frac{x-\sqrt{3}}{\sqrt{3}x+1}+\sqrt{3}}{\frac{\sqrt{3}\left(x-\sqrt{3}\right)}{\sqrt{3}x+1}-1} \\ &= -\frac{\frac{4x}{\sqrt{3}x+1}}{\frac{\sqrt{3}\left(x-\sqrt{3}\right)}{\sqrt{3}x+1}-1} \\ &= -\frac{4x}{\left(\sqrt{3}x+1\right)\left(\frac{\sqrt{3}\left(x-\sqrt{3}\right)}{\sqrt{3}x+1}-1\right)} \\ &= -\frac{4x}{\left(-\frac{4}{\sqrt{3}x+1}\right)\left(\sqrt{3}x+1\right)} \\ &= \frac{4x}{\left(x\sqrt{3}+1\right)\frac{4}{x\sqrt{3}+1}} \\ &= \frac{4x}{4} \\ &= x \end{align*}\)

\(\begin{align*} f^4(x) &= f(f^3(x)) \\ &= f(x) \\ &= \frac{x - \sqrt{3}}{x\sqrt{3} + 1} \end{align*}\)

$f^1(x) = \frac{x - \sqrt{3}}{x\sqrt{3} + 1}$
$f^2(x) = -\frac{x+\sqrt{3}}{\sqrt{3}x-1}$
$f^3(x) = x$
$f^4(x) = \frac{x - \sqrt{3}}{x\sqrt{3} + 1}$

$2021 \pmod 3 \equiv 2 \pmod 3$

$f^{2021}(x) = f^2(x) = \boxed{-\frac{x+\sqrt{3}}{\sqrt{3}x-1}}$

A(x-h)^2 +k

complete the squares  for both of em

(4-5i)(5+5i)=20+20i-25i-25(-1)

=20-5i+25

=45-5i

(11-5i)(3-2i)(2i)=(33-15i-22i+(-10))(2i)

=(33-37i-10)(2i)

=46i-74(-1)

=46i+74

https://www.wolframalpha.com/input/?i=%2811-5i%29%283-2i%29%282i%29

If you know trig: bash each angle 

if you don't: note that it is a Pythagorean triple and you coil dddfine var for the st[hottest length, set eq and solve

This is literally screaming ratios of areas.

Success = Total - Not Success

\(\begin{align*} \binom{5+7}{5} - \binom{4+3}{4}\binom{1+4}{1} &= \binom{12}{5} - \binom{7}{4}\binom{5}{1} \\ &= 792 - (35 \cdot 5) \\ &= 792 - 175 \\ &= \boxed{617} \end{align*}\)

(a) There will be a multiple of 3 if it starts with a multiple of 3.

There will be a multiple of 2 if it starts with a multiple of 2.

There will be a multiple of 5 if it starts with a multiple of 5.

There will be a multiple of 7 if it starts with a multiple of 7.

This repeats.

Thus, we only have to try (1, 3, 5), (2, 4, 6), (3, 5, 7), (5, 7, 9). The only triplet that works is (3, 5, 7).

(b) There are none. There will be a multiple of something if it gets bigger. We only try the first few, and see that they all include a number that is either 1 or composite.

Note:

$x<-4, -4 \leq x < \frac{1}{2},\frac{1}{2} \leq x<7, x\geq 7$

$x < -4$ No Solutions

$-2 < x \leq 1$ No Solutions

$1 \leq x < 14 \qquad | \qquad x = 6$

$x \geq 7$ No Solutions

$x = 6$