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12345Power functions

## Solutions to the exercises

Exercise 1
a

$a=500{p}^{-1}$

b

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.

c

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

d

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

Exercise 2
a

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

b

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.

c

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

d

$\frac{3}{\sqrt{x-1}}+5=10$ 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 $x\ge 1.36$.

Exercise 3
a

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

b

${\text{D}}_{f}=\left[3,\to ⟩$ and ${\text{B}}_{f}=\left[-5,\to ⟩$
${\text{D}}_{g}=\left[0,\to ⟩$ and ${\text{B}}_{g}=\left[0,\to ⟩$

c

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

Exercise 4
a

$f\left(x\right)=110{\left(x-10\right)}^{-2}+25$ 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.

b

$x=10$ and $y=25$

c

${\text{D}}_{f}=⟨←,10⟩\cup ⟨10,\to ⟩$ and ${\text{B}}_{f}=⟨25,\to ⟩$

d

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

Exercise 5
a

When $c$ is an even whole number.

b

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

c

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