Rom

$$y=\dfrac{2x+1}{3x-2}$$

First off note that this isn't defined at x=2/3 but other than that it is one to one.

$$(3x-2)y=2x+1$$

$$3xy - 2y = 2x+1$$

$$x(3y-2)=1+2y$$

$$x=\dfrac {1+2y}{3y-2}$$

now just replace x by f^-1(x) and y by x to obtain

$$f^{-1}(x)=\dfrac{2x+1}{3x-2}~~\forall x \neq \dfrac 2 3$$

For real numbers log(x) function isn't defined for x<=0. So for this to be undefined

$$8x^2+26x-7<0$$

$$4x-1<0$$

and further division by 0 isn't permitted so

$$\log(4x-1)= 0 \Rightarrow (4x-1)= 1$$

Sorting through this we see

$$(2x+7)(4x-1)\leq 0 \Rightarrow \left[-\dfrac 7 2 \leq x \leq \dfrac 1 4 \right]$$

This also takes care of the second condition.

We also have

$$4x-1 = 1 \Rightarrow 4x = 2 \Rightarrow x = \dfrac 1 2$$

Combing these we have that the original function is undefined on

$$\left[-\dfrac 7 2 \leq x \leq \dfrac 1 4 \right] \bigcup \left\{\dfrac 1 2 \right\}$$

(2x+3)(x-1) > 0 for all x > 1 thus k=0

$${\left({\mathtt{1\,406.25}}\right)}^{{\mathtt{3}}} = {\mathtt{2\,780\,914\,306.640\: \!625}}$$

$$\sin^2(x)+\cos^2(x)=1$$

$$\cos(x)=\sqrt{1-\sin^2(x)}$$

$$\cos(x)=\sqrt{1-(0.76)^2}$$

$${\sqrt{{\mathtt{1}}{\mathtt{\,-\,}}{\left({\mathtt{0.76}}\right)}^{{\mathtt{2}}}}} = {\mathtt{0.649\: \!923\: \!072\: \!370\: \!876\: \!8}}$$

$$\cos(x) \approx 0.65$$

$$\sin^2(x)+\cos^2(x)=1$$

$$\cos(x)=\sqrt{1-\sin^2(x)}$$

$$\cos(x)=\sqrt{1-(0.76)^2}$$

$${\sqrt{{\mathtt{1}}{\mathtt{\,-\,}}{\left({\mathtt{0.76}}\right)}^{{\mathtt{2}}}}} = {\mathtt{0.649\: \!923\: \!072\: \!370\: \!876\: \!8}}$$

$$\cos(x) \approx 0.65$$

$${\frac{{\mathtt{1\,270}}}{{\mathtt{3}}}} = {\mathtt{423.333\: \!333\: \!333\: \!333\: \!333\: \!3}}$$

$$x(x+29)=1750$$

$$x^2+29x-1750=0$$

now use the quadratic formula to find the values of x that satisfy this

a=1

b=29

c=-1750

$$r1,r2=\dfrac{-b\pm \sqrt{b^2-4 a c}}{2 a}$$

$${\frac{\left({\mathtt{\,-\,}}{\mathtt{29}}{\mathtt{\,\small\textbf+\,}}{\sqrt{{{\mathtt{29}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{4}}{\mathtt{\,\times\,}}{\mathtt{1}}{\mathtt{\,\times\,}}\left(-{\mathtt{1\,750}}\right)}}\right)}{\left({\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{1}}\right)}} = {\mathtt{29.774\: \!710\: \!614\: \!525\: \!76}}$$

$${\frac{\left({\mathtt{\,-\,}}{\mathtt{29}}{\mathtt{\,-\,}}{\sqrt{{{\mathtt{29}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{4}}{\mathtt{\,\times\,}}{\mathtt{1}}{\mathtt{\,\times\,}}\left(-{\mathtt{1\,750}}\right)}}\right)}{\left({\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{1}}\right)}} = -{\mathtt{58.774\: \!710\: \!614\: \!525\: \!76}}$$

and these are the two values of x that satisfy the original equation

$${\mathtt{29.774\: \!7}}{\mathtt{\,\times\,}}\left({\mathtt{29.774\: \!7}}{\mathtt{\,\small\textbf+\,}}{\mathtt{29}}\right) = {\mathtt{1\,749.999\: \!060\: \!09}}$$

$$\left(-{\mathtt{58.774\: \!7}}{\mathtt{\,\times\,}}\left({\mathtt{\,-\,}}{\mathtt{58.774\: \!7}}{\mathtt{\,\small\textbf+\,}}{\mathtt{29}}\right)\right) = {\mathtt{1\,749.999\: \!060\: \!09}}$$

the cosine is the ratio of the adjacent leg of it's angle to the hypotenuse

$$\cos(\theta)=\dfrac {24}{45}$$

$$\theta = \arccos\left(\dfrac{24}{45}\right)$$

$$\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{cos}}^{\!\!\mathtt{-1}}{\left({\frac{{\mathtt{24}}}{{\mathtt{45}}}}\right)} = {\mathtt{57.769\: \!047\: \!364\: \!498^{\circ}}}$$

$$1 \text{ picometer is }10^{-12} \text{ meters}$$

$$506 \text{ picometers is }506 \times 10^{-12}=5.06\times 10^{-10}\text { meters}$$