Function a. $y=\sqrt{x}-2$ has a domain of $\left\{x\mid x\ge 0\mathrm{,}\mathrm{x}\in \mathrm{R}\right\}$ because only $x$ is under the root sign.

Range:

As $x$ increases in value, so does $y$. As such, the smallest possible value of $y$ corresponds to the smallest possible value of $x$, which in this case is zero.


The range of function a. is $\left\{y\mid y\ge -2\mathrm{,}\mathrm{y}\in \mathrm{R}\right\}$.

Function a. matches graph B.

The function b. $y=\sqrt{-x}+2$ has a domain of $\left\{x\mid x\le 0\mathrm{,}\mathrm{x}\in \mathrm{R}\right\}$ because $-x$ is under the square root sign. The value under the root sign can be $0$ or any positive number. If $x$ is negative, then it is a negative multiplied by a negative under the root sign, which results in a positive value.


Solve for $x$.


which reads, "$0$ is greater than or equal to $x\dots$ or $x$ is less than or equal to $0$.”

Alternatively,


Solve for $x$ by dividing both sides by $-1$.


When dividing both sides of an inequality by a negative value, flip the inequality sign.


The range can be determined using the domain.

Start with $x=0$.


Then, try another value from the domain.


As $x$ decreases, $y$ increases.

The range of function b. is $\left\{y\mid y\ge 2\mathrm{,}\mathrm{y}\in \mathrm{R}\right\}$.

The function b. matches graph A.

Function c. $y=-\sqrt{x+2}$ has a domain of $\left\{x\mid x\ge -2\mathrm{,}\mathrm{x}\in \mathrm{R}\right\}$.


Use the domain to determine the range. Use the first value of the domain, $x=-2$.


Then, use another value from the domain, perhaps $x=-1$, to determine the direction of the range.


As $x$ increases, $y$ decreases. The range of function c. is $\left\{y\mid y\le 0\mathrm{,}\mathrm{y}\in \mathrm{R}\right\}$.

Function c. matches graph D.

Function d. $y=-\sqrt{-\left(x-2\right)}$ has a domain of $\left\{x\mid x\le 2\mathrm{,}\mathrm{x}\in \mathrm{R}\right\}$.


or $x\le 2$

Use the domain to determine the range. Use the first value of the domain, $x=2$.


Then, use another value from the domain, perhaps $x=1$, to determine the direction of the range.


As $x$ decreases, $y$ decreases.

The range of function d. is $\left\{y\mid y\le 0\mathrm{,}\mathrm{y}\in \mathrm{R}\right\}$.

Function d. matches graph C.

Domain: The domain is the set of permissible values for the variable $d$, or the number of days before the release date.

$-d$ is under the square root sign; therefore, $-d\ge 0$.

Add $d$ to both sides of the inequality.


The value of $d$ must be less than or equal to $0$.

Range: Use $d=0$ in the function $P\left(d\right)$.


Now, use another domain value to determine the direction of the range.


As $d$ decreases, $P$ decreases from $20$ million. Because $P$ represents the number of pre-orders, it must be greater than or equal to zero.


In this situation, the domain represents the number of days prior to a release date. It is a countdown, or the amount of time before the event happens. It is negative because it represents the difference between a future date and the date of the pre-orders.

In this situation, the range represents the number of pre-orders, in millions.

As time draws closer to the release date, the number of orders increases to $20000000$. It is not possible to have fewer than $0$ pre-orders.

The range cannot be less than zero, so the domain needs to include a restriction to reflect this.

The largest domain value is $d=0$. It corresponds to the largest range value of $20$ million.

Use $P\left(d\right)=0$, the smallest range value, to determine the largest possible domain value.


Square both sides of the equation. Then, solve for $d$.


Therefore, the domain for this situation is $\left\{d\left|-100\le d\le 0\mathrm{,}d\in \mathrm{R}\right.\right\}$, which means that there will be $100$ days of pre-sales prior to the date of release.