Solve the inequalities: 1/(x+6) < 2/3-3n. Find domain and range of functions g(x)=x−2, h(x)=7n/(x²-f), k(x)=(x+1)/sqrt(2x+1). Find the range of values of x for which |2x-4| <= 4.

As $y$ goes from 0 to $\infty$, $x$ goes from 0 to $\infty$.

Range: $[0,\infty)$

[0,\infty)

1(b) $g(x)=2^{x}$

Domain:

Exponential defined for all real $x$.

Range:

As $x\to -\infty$, $y\to 0^+$; as $x\to \infty$, $y\to \infty$.

Range: $(0,\infty)$

(0,\infty)

2(a) $m(x)=x^2$

Domain: $\mathbb{R}$ (polynomial).

Range:

For all $y\ge 0$, real $x$ exists.

Minimum $y=0$ at $x=0$; $y\to\infty$ as $|x|\to\infty$.

Range: $[0,\infty)$

[0,\infty)

2(b) $n(x)=x^3$

Domain: $\mathbb{R}$.

Range:

Bijective from $\mathbb{R}$ to $\mathbb{R}$.

Range: $(-\infty,\infty)$

(-\infty,\infty)

3. $u(x)=\sqrt{x^2-4}$

Domain:

Inside square root $\ge 0$.

$x\le -2$ or $x\ge 2$

Range:

Always true for $y\ge 0$.

Minimum $y=0$ at $|x|=2$; $y\to\infty$ as $|x|\to\infty$.

Range: $[0,\infty)$

[0,\infty)

4. $v(x)=\ln(x^2-1)$

Domain:

Argument $>0$.

Range:

$x^2=e^y+1>1$ (since $e^y>0$)

As $|x|\to 1^+$, $x^2-1\to 0^+$, $y\to -\infty$.

As $|x|\to\infty$, $y\to\infty$.

Range: $(-\infty,\infty)$

(-\infty,\infty)