Figure

EXAMPLE 1Forming the Sum, Difference, Product, and Quotient of Two Functions

Let $f$ and $g$ be two functions defined as $f( x) =\sqrt{x-1}\qquad\hbox{and}\qquad g( x) =\sqrt{4-x}$ Find the following functions and determine their domain:

$( f+g) ( x)$
$(f-g) ( x)$
$(f\cdot g)( x)$
$\left( \dfrac{f}{g}\!\right) ( x)$

The domain of $f$ is $\{ x|x\geq 1\}$ , and the domain of $g$ is $\{ x |x \leq 4\}$ .

$( f+g) ( x) =f( x) +g( x) =\sqrt{x-1}+\sqrt{4-x}$ . The domain of $( f+g) ( x)$ is the closed interval $[ 1,4]$ .
$( f-g) ( x) =f( x) -g( x) =\sqrt{x-1}-\sqrt{4-x}$ . The domain of $( f-g) ( x)$ is the closed interval $[ 1,4]$ .
$( f\cdot g) ( x) =f( x) \cdot g( x)\, {=}\,( \sqrt{x-1}) ( \sqrt{4-x} ) =\sqrt{-x^{2}+5x-4}$ . The domain of $( f\cdot g) ( x)$ is the closed interval $[ 1,4]$ .
$\left( \dfrac{f}{g}\right)\ ( x) =\dfrac{ f( x) }{g( x) }=\dfrac{\sqrt{x-1}}{\sqrt{4-x}}=\dfrac{ \sqrt{-x^{2}+5x-4}}{4-x}$ . The domain of $\left( \dfrac{f}{g}\right) ( x)$ is the half-open interval $[1,4)$ .