$a=500p^{-1}$

If $p=2.50$, then $a=200$ and if $p=5.00$, then $a=100$. If you double the price then sales are exactly halved.

If $a=300$, then $p=\frac{500}{300}\approx 1.67$. Formula: $p=\frac{500}{a}$.

If $p=0.01$, then $a=50000$ and if $p=100$, then $a=5$. Therefore $0.50\le p\le 5$.

$f\left(x\right)=3\cdot {(x-1)}^{\mathrm{-}\frac{1}{2}}+5$

First you shift the graph by $1$ in the $x$ -direction, then you multiply by $3$ in the $y$ -direction, and finally you shift the graph by $5$ in the $y$ -direction.

${\text{D}}_f=⟨1,\to ⟩$ and ${\text{B}}_f=⟨5,\to ⟩$

$\frac{3}{\sqrt{x-1}}+5=10$ gives: $\frac{3}{\sqrt{x-1}}=5$ and $\sqrt{x-1}=0.6$, so that $x=1.36$.
$f\left(x\right)\le 10$ if $x\ge 1.36$.

$f\left(x\right)=-5+2{(x-3)}^{\frac{1}{2}}$ and $g\left(x\right)=x^{\frac{1}{2}}$.
You first shift $3$ units in the $x$ -direction, then you multiply by $2$ along the $x$ -axis, and you finally shift $-5$ units in the $y$ -direction.

${\text{D}}_f=[3,\to ⟩$ and ${\text{B}}_f=[-5,\to ⟩$
${\text{D}}_g=[0,\to ⟩$ and ${\text{B}}_g=[0,\to ⟩$

$-5+2{(x-3)}^{\frac{1}{2}}=100$ gives ${(x-3)}^{\frac{1}{2}}=52.5$ and therefore $x=2759.25$.
$f\left(x\right)\ge 100$ for $x\ge 2759.25$.

$f\left(x\right)=110{(x-10)}^{-2}+25$ is derived from $y=x^{-2}$ by: shifting $10$ units in the $x$ -direction, multiplying by $100$ with respect to the $x$ -axis and shifting $25$ units in the $y$ -direction.

$x=10$ and $y=25$

${\text{D}}_f=⟨\leftarrow ,10⟩\cup ⟨10,\to ⟩$ and ${\text{B}}_f=⟨25,\to ⟩$

$f\left(x\right)=50$ gives you ${(x-10)}^2=4$ and $x=8\vee x=12$.
$f\left(x\right)\le 50$ for $x\le 8\vee x\ge 12$.

When $c$ is an even whole number.

If the value of $a$ is positive (minimum) or negative (maximum).

$b$ and $d$ indicate the shifts with respect to the basic function. The coordinates of the vertex are $(b,d)$.