what is the domain, range for these equations? kindly include explaination.
1. z = 4x² + y²
2. z = √(x² + y² - 16)
3. z = √(4 - y²)
4. z = 1/x²y²
Hello, fuzzyorama!
$\displaystyle x$ can be any real value.What is the domain, range for these equations?
$\displaystyle 1)\; z\:=\:4x^2 + y^2$
$\displaystyle y$ can be any real value.
. . Domain: .$\displaystyle x,y \in (-\infty,\,\infty)$
Since $\displaystyle x^2$ and $\displaystyle y^2$ are nonnegative, $\displaystyle z$ will be nonnegative.
. . Range: .$\displaystyle z \,\geq\,0$
$\displaystyle 2)\; z \:= \:\sqrt{x^2 + y^2 - 16}$
The radicand must be nonnegative.
. . Domain: .$\displaystyle x^2+y^2\,\geq\,16$
$\displaystyle z$ will be nonnegative.
. . Range: .$\displaystyle z \,\geq\,0$
$\displaystyle 3)\; z \:= \:\sqrt{4 - y^2}$
The radicand must be nonegative. .$\displaystyle 4-y^2\:\geq\:0$
. . Domain: .$\displaystyle -2\:\leq \:y\:\leq\:2$
$\displaystyle z$ will be nonnegative.
. . Range: .$\displaystyle z\,\geq\,0$
$\displaystyle 4)\; z \:= \:\frac{1}{x^2y^2}$
$\displaystyle x,\,y$ cannot be zero.
. . Domain: .$\displaystyle x,y\:\neq\:0$
Since $\displaystyle x^2y^2$ is always positive, $\displaystyle z$ is positive.
. . Range: .$\displaystyle z\,>\,0$