Range of a function and images of sets

$f(x) = 2 x-5, x \in [-2 ; 2]$

$f(x) = \left|x-1\right|, x \in [0 ; 5]$

$f(x) = x+\operatorname {sign} x, x \in \mathbb {R}$

$f(x) = x^{2}+2 x-3, x \in \mathbb {R}$

$f(x) = -2 x^{2}+x+1, x \in \mathbb {R}$

$f(x) = 5-12 x-2 x^{2}, x \in [-4 ; 1]$;

$f(x) = x+\frac{1}{x}, x \in (0 ;+\infty )$;

$f(x) = \frac{x^{2}+4}{x}, x \in (-\infty ; 0)$;

$f(x) = \frac{x^{2}}{x^{2}+1}, x \in \mathbb {R}$.

$y = \sqrt{x^{2}+1}$

$y = \sqrt[4]{x^{2}-1}$

$y = \sqrt{x(4-x)}$;

$y = \sqrt{\frac{9 x^{2}+1}{x}}$;

$y = a x+\frac{b}{x}$, where $a b > 0$;

$y = a x+\frac{b}{x}$, where $a b < 0$;

$y = \frac{x^{2}+2 x-2}{x^{2}-x+1}$.

$y = 10^{-x^{2}}$

$y = \frac{1}{1-2^{-x}}$

$y = 4^{x}-2^{x}+1$;

$y = \lg \left(x^{2}+10\right)$

$y = \log_{2}\left(4-x^{4}\right)$;

$y = \log_{3} x+\log_{x} 3$.

$y = 1-2\left|\cos x\right|$

$y = \sin x+\sin (x+\pi / 3)$;

$y = \sin^{4} x+\cos^{4} x$;

$y = \frac{1+\sin x}{\sin x}$

$y = \arccos \left|x\right|$;

$y = \pi -\left|\operatorname {arctg} x\right|$;

$y = \cos (\arcsin x)$

$y = \operatorname {arctg} \frac{2 x}{x^{2}+1}$.

$y = \frac{2 x}{x^{2}+9}$

$y = \frac{3-x^{2}}{3+x^{2}}$

$y = \sqrt{x^{2}+2 x+2}$

$y = \sqrt{8-2 x-x^{2}}$;

$y = \sqrt{2 x-1-x^{2}}$;

$y = 2^{x^{2}+4 x-5}$

$y = \log_{3}\left(5+4 x-x^{2}\right)$;

$y = \log_{x^{2}} x$;

$y = \sqrt{2 \log_{2} x-\log_{2}^{2} x}$.

$y = \sin x-5 \cos x$;

$y = 1-2\left|\sin 2 x\right|$;

$y = \frac{1+\cos x}{1-\cos x}$

$y = \frac{\sin x-\cos x}{\sin x+\cos x}$;

$y = \sqrt{\lg \sin x}$;

$y = \cos^{2} x-\sin x$;

$y = \log_{2}\left(\cos x+\sin^{2} x\right)$;

$y = \cos \left(\frac{1}{2} \arcsin x\right)$;

$y = \operatorname {arcctg}(\sin x)$.

Prove that for any $x \in \mathbb {R}$ : а) $\mathbb {E}\left[x+1\right] = \mathbb {E}\left[x\right]+1$ б) $\mathbb {E}\left[x+n\right] = \mathbb {E}\left[x\right]+n, n \in \mathbb {Z}$.

Find the range of the function $x-\mathbb {E}\left[x-2\right]$.

$f(x) = \sqrt{\frac{2 a+x}{a-x}}, \quad g(x) = \frac{1}{x^{2}+2 x+4 a-2}$

$f(x) = \lg \left(x^{2}+a\right), \quad g(x) = \frac{x^{2}+x-a}{x}$;

$f(x) = \arcsin \left(2^{x}-a\right), \quad g(x) = \log_{2}\left(2 a+1 / 2-a \cdot 2^{x}\right)$.