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.

1(a) $f(x)=\sqrt{x}$

Domain:

Square root requires non-negative argument.

$x\ge 0$

Domain: $[0,\infty)$

Range:

Step 1: Let $y=f(x)$.

$y=\sqrt{x}$

Step 2: Square both sides.

$y^2=x$

Step 3: $y\ge 0$ (principal square root).

Step 4: $x=y^2\ge 0$, which is in domain for all $y\ge 0$.

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 $x$.

$D=(-\infty,\infty)$

Range:

Step 1: Let $y=g(x)$.

$y=2^{x}$

Step 2: $y>0$ always since $2^x>0$.

Step 3: $x=\log_2 y$, defined for $y>0$.

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:

Step 1: Let $y=m(x)$.

$y=x^2$

Step 2: $x=\pm \sqrt{y}$.

Step 3: Real $x$ requires $y\ge 0$.

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:

Step 1: Let $y=n(x)$.

$y=x^3$

Step 2: $x=\sqrt[3]{y}$.

Step 3: Cube root defined for all real $y$.

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

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

(-\infty,\infty)

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

Domain:

Inside square root $\ge 0$.

$x^2-4\ge 0$

$x^2\ge 4$

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

Range:

Step 1: Let $y=u(x)$.

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

$y\ge 0$

Step 2: Square both sides.

$y^2=x^2-4$

$x^2=y^2+4$

Step 3: $|x|=\sqrt{y^2+4}\ge 2$, since $y^2+4\ge 4$.

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 $>0$.

$x^2-1>0$

$|x|>1$

Range:

Step 1: Let $y=v(x)$.

$y=\ln(x^2-1)$

Step 2: $x^2-1=e^y$

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

Step 3: Always true for all real $y$.

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

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

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

(-\infty,\infty)